To be frank, this is very improbable, and in the end, wouldn't change much. To fully grasp the position Poland found itself in the aftermath of the Great War, you have to know and understand what came before. Following its partition in the latter half of the 18th century, there would be numerous revolts and insurrections, such as the January Uprising, November Uprising, and Greater Poland Uprising (as part of the revolutions of 1848). Due to the unsuccessful nature of all these revolts, more and more Poles would come to view armed resistance against the partitioning powers futile, and that rather, the only path to an independent Polish state must be through a program of 'organic work', which would encourage basic things such as hygiene and education, and that through things such as this, nation building and national revival could occur. This method would be encouraged by many positivist writers and artists. Additionally, more and more Poles would become skeptical of their past, with modern Darwinism cause them to rather view the state a failure, with its multiculturalism, toleration and inclusiveness, one of the leading causes of its downfall.
Going into the Great War, there would be two main camps, Roman Dmowski, and his National Democrats, as well as Jozef Pilsudski, and his Polish Peoples Party. The National Democrats had a system of beliefs best described by Dmowski's book Thoughts of a Modern Pole, in which he describes a state which would assimilate and embrace all those willing, which he described as a 'healthy national egoism'. However, all minorities unwilling to assimilate would be viewed as alien bodies within the nation, and their expulsion necessary to ensure the success of the state. To the contrary, Pilsudski and his socialists, unlike the National Democrats, who disavowed traditional Polish values such as multiculturalism and toleration, sought out to restore the commonwealth's former frontiers dating back to 1772. To make matters worse, there was no conservative political organization to balance out these movements, as most conservatives acquiesced in the status quo and refrained from subversive politics.
By the time of the signing of the Armistice on August 11th, the Poles had several provisional governments established. One was Roman Dmowski’s National Committee, recognized by the French, and established in Paris a year prior. Another was the Lublin Provisional Government, which had formed a mere day before the signing of the armistice, headed by the socialist Ignacy Daszyński. Not only this, but there was also the Regency Council of the Kingdom of Poland, a planned German satellite state. However, On June 10th, a day before the Armistice, Pilsudski, having been arrested during the oath crisis, would be released and arrive in Warsaw, where he would be greeted by the Regency council, who would turn over their powers of state to him, with him declaring “The Polish State has arisen from the will of the whole nation.” This alone concerned the Entente, having a government waiting in Paris ready to be installed, would have their efforts somewhat subverted, for now, a rather unpredictable political operative, who had fought under the central powers, was taking the reigns in the formation of the Polish state. Not to be outdone, a compromise would be reached, with Paderewski arriving in Polish to serve as Prime minister in a coalition government with Pilsudski serving as head of state.
In regards to actually convincing the former peoples of the commonwealth to join in some form of federation with Poland, there was serious contention, between nationalists of all prospective member states. The Lithuanians, having taken part in the failed 1863 January uprising, would come to view any potential alliance with the Poles worthless, and also lay claim to the territories of the former Grand Duchy of Lithuania, including the city of Wilno, which while being their historical capital, only two percent of its inhabitants spoke Lithuanian. There would also be the issue of Ukrainian nationalists, who believe in the legacy of the Kievan Rus (and as a result the city of Lwow, where Poles outnumbered Ukrainians with a ratio of two to one) which would, in turn, be disputed by Russian nationalists, who viewed Ukrainian simply as a dialect of Russian.
Too many conflicts would exist with Poland's neighbours, as well as Pilsudski's domestic political opponents, to make the formation of intermarium a realisable goal. With Poland's invasion of the Ukraine, Pilsudski would form an alliance with the leader of the Ukrainian People's Republic, Petliura, who had been ousted from his seat of government in Kiev by Bolshevik forces and supporters. This alliance would expend much of Pilsudski's political capital, and would be met with significant internal opposition in Poland and Ukraine. Lack of Ukrainian Support would also be exemplified, not only by their failure to hold their capital, but at the peak of the Kiev Offensive, the Ukrainians would only come to bear two, understrength infantry divisions, insufficient to say the least. This showed there simply wasn't enough support for independence, let alone participation in a Polish-led federation, for many Ukrainians weren't supportive of the Polish offensive campaign and simply viewed them as invaders rather than liberators.
The situation with Lithuania wasn't much better. While Dmowski and the national democrats supported the idea of a union with Lithuania (argued for by Dmowski, who had been a key member of the Polish delegation, at the Paris peace conference, likely with intention to assimilate them), and Pilsudski's Polish Military Organisation had planned a coup, the idea of union would not come to fruition and the planned coup's plotters being arrested by Lithuanian intelligence services. What made matters worse was the contentious position of the city of Wilno, which would eventually be transferred to Poland following Zeligowski's mutiny and the short-lived Central Lithuanian Republic. This would result in cold relations between Poland and Lithuania for the rest of the interwar period.
So, let's say Poland achieves a union with Lithuania, and triumphs over the Bolsheviks in the war, what happens then? Will Pilsudski resign as chief of state? Will he still launch his coup in 1926? Likely. For intermarium to have the best outcome, Pilsudski (it being his vision and all) must take the helm. A key factor in the degree of success of intermarium comes from the sheer amount of land split off from the Ukraine. It is likely that unrest within Ukrainian territories would be a thing, at least at first. The historical formation of the Border Protection Corps would likely be sooner, to ensure stability within its eastern border regions. With the loss of the Ukraine, the five-year plans of Stalin would likely be far less effective (once again, dependent on how much of Ukraine would be taken, the most plausible being Kiev and westwards) The result, unfortunately, is still gloomy. The Soviets will now have a far greater axe to grind with the Poles, and the Poles would now have a far larger population of questionable loyalty within their borders than they had historically. While the Poles could now utilise Ukrainian wheat and other resources to fuel modernisation of their armed forces, as well as hopefully a larger potential pool of recruits, the degree of success would once again matter on the amount of land acquired. The amount of autonomy granted to this region would also be influential: would they have their own service branch, a sort of independent, national guard? or would they simply have units integrated within the Polish military, and if that, would they have Polish or Ukrainian officers, etc. Pilsudski would likely implement progressive policies as he did historically in regards to treatment of Ukrainians, and the same would be for the Lithuanians and any Byelorussians. For Pilsudski and his Sanacja government to truly maximise the potential of this territory, they would have to exploit it, even if not to the extent of a 5-year plan. This, especially under Pilsudski's sanacja government, would be extremely unlikely.
Let's say Pilsudski never launches his coup, or it fails. The National Democrats or whoever else in government would likely have a repressive stance toward the sizable Ukrainian as well as the Byelorussian minorities, and a softer one towards the Lithuanians. Attempts of Polanisation would be all but guaranteed, and exploitation would likely accompany efforts. Industrial expansion efforts would be prioritised, for Polish territories, having been overlooked by the partitioning powers, would be far behind their neighbours in term of potential. Even with an autonomous Ukraine and Lithuania, or White Russia, the Poles would dominate the Federation and attempt to extort or tax them to fuel modernisation and industrial reform within their own territory.
So as time progress, going into the thirties, Intermarium and Poland would have greater power than they had historically (though again, dependent on territory gained). Pilsudski would continue to posture militarily, flexing his muscles, however any preemptive measures against Germany, apart from a more impactful trade-war, would be unlikely. Pilsudski, and his successors, would most likely not act alone, and the western allies would likely show disinterest in such an idea as they had historically. However, assuming Pilsudski dies in 1935 as he did historically, his Sanacja government and successor colonel's regime would likely follow the same path it did historically, being that of reversing Pilsudski's progressive policies of treatment towards minorities, as well as enacting repressive ones. So with Pilsudski dead, or him having never seized control of the government, either long-term or short-term repression of minorities would, I imagine, result in some form of Ukrainian and Byelorussian insurrection in the late thirties, no doubt with the vengeful Soviet Union actively fueling anti-Polish sentiment and arming insurrectionists, as well as launching cross-border raids as they did historically but continuing into the thirties.
By lets say 1939, Poland modernisation of their armed forces would likely be more successful, though, as always, dependent on territory acquired. Unfortunately, open revolt or rebellion in the eastern-border regions would likely be the culmination of the civil unrest which had been brewing due to the repressive government policies, attempts of Polanisation, and exploitation of the territory. Ukrainian members of the armed forces, whether an independent branch or integrated formation, would likely join in the revolt. Predictably, the Soviets would supply these rebels with arms and volunteers, if not using it as justification to invade or initiate a border war (similar to that seen in Manchuria) in protection of proles being oppressed by the capitalist Poles.
How a stronger Poland would have affected Germany posturing and expansion would also be interesting. Unfortunately, it is likely the western allies would have maintained their policy of appeasement, which would begin as early as 1926, preferring warmed relations rather than strained ones with Germany, even if there was a stronger counterbalance to them in Eastern Europe in the form of Intermarium / Poland. One thing I can picture changing is the German ultimatum for Memel, in which case, I doubt they would even send it, for they never challenged Poland historically as early as 1938, let alone a strengthened one.
Caught fighting a war in the eastern borderlands, and with large minorities of dubious loyalties in both the west and east, Poland would find itself once more in a precarious position, even with a strengthened military, they would likely succumb to a joint German-Soviet invasion as they had historically, albeit showing hopefully stiffer resistance and lasting longer before capitulating.
edit: I reccommend reading Adam Zamoyski's Poland: A History to better grasp the circumstances
Going into the Great War, there would be two main camps, Roman Dmowski, and his National Democrats, as well as Jozef Pilsudski, and his Polish Peoples Party. The National Democrats had a system of beliefs best described by Dmowski's book Thoughts of a Modern Pole, in which he describes a state which would assimilate and embrace all those willing, which he described as a 'healthy national egoism'. However, all minorities unwilling to assimilate would be viewed as alien bodies within the nation, and their expulsion necessary to ensure the success of the state. To the contrary, Pilsudski and his socialists, unlike the National Democrats, who disavowed traditional Polish values such as multiculturalism and toleration, sought out to restore the commonwealth's former frontiers dating back to 1772. To make matters worse, there was no conservative political organization to balance out these movements, as most conservatives acquiesced in the status quo and refrained from subversive politics.
By the time of the signing of the Armistice on August 11th, the Poles had several provisional governments established. One was Roman Dmowski’s National Committee, recognized by the French, and established in Paris a year prior. Another was the Lublin Provisional Government, which had formed a mere day before the signing of the armistice, headed by the socialist Ignacy Daszyński. Not only this, but there was also the Regency Council of the Kingdom of Poland, a planned German satellite state. However, On June 10th, a day before the Armistice, Pilsudski, having been arrested during the oath crisis, would be released and arrive in Warsaw, where he would be greeted by the Regency council, who would turn over their powers of state to him, with him declaring “The Polish State has arisen from the will of the whole nation.” This alone concerned the Entente, having a government waiting in Paris ready to be installed, would have their efforts somewhat subverted, for now, a rather unpredictable political operative, who had fought under the central powers, was taking the reigns in the formation of the Polish state. Not to be outdone, a compromise would be reached, with Paderewski arriving in Polish to serve as Prime minister in a coalition government with Pilsudski serving as head of state.
In regards to actually convincing the former peoples of the commonwealth to join in some form of federation with Poland, there was serious contention, between nationalists of all prospective member states. The Lithuanians, having taken part in the failed 1863 January uprising, would come to view any potential alliance with the Poles worthless, and also lay claim to the territories of the former Grand Duchy of Lithuania, including the city of Wilno, which while being their historical capital, only two percent of its inhabitants spoke Lithuanian. There would also be the issue of Ukrainian nationalists, who believe in the legacy of the Kievan Rus (and as a result the city of Lwow, where Poles outnumbered Ukrainians with a ratio of two to one) which would, in turn, be disputed by Russian nationalists, who viewed Ukrainian simply as a dialect of Russian.
Too many conflicts would exist with Poland's neighbours, as well as Pilsudski's domestic political opponents, to make the formation of intermarium a realisable goal. With Poland's invasion of the Ukraine, Pilsudski would form an alliance with the leader of the Ukrainian People's Republic, Petliura, who had been ousted from his seat of government in Kiev by Bolshevik forces and supporters. This alliance would expend much of Pilsudski's political capital, and would be met with significant internal opposition in Poland and Ukraine. Lack of Ukrainian Support would also be exemplified, not only by their failure to hold their capital, but at the peak of the Kiev Offensive, the Ukrainians would only come to bear two, understrength infantry divisions, insufficient to say the least. This showed there simply wasn't enough support for independence, let alone participation in a Polish-led federation, for many Ukrainians weren't supportive of the Polish offensive campaign and simply viewed them as invaders rather than liberators.
The situation with Lithuania wasn't much better. While Dmowski and the national democrats supported the idea of a union with Lithuania (argued for by Dmowski, who had been a key member of the Polish delegation, at the Paris peace conference, likely with intention to assimilate them), and Pilsudski's Polish Military Organisation had planned a coup, the idea of union would not come to fruition and the planned coup's plotters being arrested by Lithuanian intelligence services. What made matters worse was the contentious position of the city of Wilno, which would eventually be transferred to Poland following Zeligowski's mutiny and the short-lived Central Lithuanian Republic. This would result in cold relations between Poland and Lithuania for the rest of the interwar period.
So, let's say Poland achieves a union with Lithuania, and triumphs over the Bolsheviks in the war, what happens then? Will Pilsudski resign as chief of state? Will he still launch his coup in 1926? Likely. For intermarium to have the best outcome, Pilsudski (it being his vision and all) must take the helm. A key factor in the degree of success of intermarium comes from the sheer amount of land split off from the Ukraine. It is likely that unrest within Ukrainian territories would be a thing, at least at first. The historical formation of the Border Protection Corps would likely be sooner, to ensure stability within its eastern border regions. With the loss of the Ukraine, the five-year plans of Stalin would likely be far less effective (once again, dependent on how much of Ukraine would be taken, the most plausible being Kiev and westwards) The result, unfortunately, is still gloomy. The Soviets will now have a far greater axe to grind with the Poles, and the Poles would now have a far larger population of questionable loyalty within their borders than they had historically. While the Poles could now utilise Ukrainian wheat and other resources to fuel modernisation of their armed forces, as well as hopefully a larger potential pool of recruits, the degree of success would once again matter on the amount of land acquired. The amount of autonomy granted to this region would also be influential: would they have their own service branch, a sort of independent, national guard? or would they simply have units integrated within the Polish military, and if that, would they have Polish or Ukrainian officers, etc. Pilsudski would likely implement progressive policies as he did historically in regards to treatment of Ukrainians, and the same would be for the Lithuanians and any Byelorussians. For Pilsudski and his Sanacja government to truly maximise the potential of this territory, they would have to exploit it, even if not to the extent of a 5-year plan. This, especially under Pilsudski's sanacja government, would be extremely unlikely.
Let's say Pilsudski never launches his coup, or it fails. The National Democrats or whoever else in government would likely have a repressive stance toward the sizable Ukrainian as well as the Byelorussian minorities, and a softer one towards the Lithuanians. Attempts of Polanisation would be all but guaranteed, and exploitation would likely accompany efforts. Industrial expansion efforts would be prioritised, for Polish territories, having been overlooked by the partitioning powers, would be far behind their neighbours in term of potential. Even with an autonomous Ukraine and Lithuania, or White Russia, the Poles would dominate the Federation and attempt to extort or tax them to fuel modernisation and industrial reform within their own territory.
So as time progress, going into the thirties, Intermarium and Poland would have greater power than they had historically (though again, dependent on territory gained). Pilsudski would continue to posture militarily, flexing his muscles, however any preemptive measures against Germany, apart from a more impactful trade-war, would be unlikely. Pilsudski, and his successors, would most likely not act alone, and the western allies would likely show disinterest in such an idea as they had historically. However, assuming Pilsudski dies in 1935 as he did historically, his Sanacja government and successor colonel's regime would likely follow the same path it did historically, being that of reversing Pilsudski's progressive policies of treatment towards minorities, as well as enacting repressive ones. So with Pilsudski dead, or him having never seized control of the government, either long-term or short-term repression of minorities would, I imagine, result in some form of Ukrainian and Byelorussian insurrection in the late thirties, no doubt with the vengeful Soviet Union actively fueling anti-Polish sentiment and arming insurrectionists, as well as launching cross-border raids as they did historically but continuing into the thirties.
By lets say 1939, Poland modernisation of their armed forces would likely be more successful, though, as always, dependent on territory acquired. Unfortunately, open revolt or rebellion in the eastern-border regions would likely be the culmination of the civil unrest which had been brewing due to the repressive government policies, attempts of Polanisation, and exploitation of the territory. Ukrainian members of the armed forces, whether an independent branch or integrated formation, would likely join in the revolt. Predictably, the Soviets would supply these rebels with arms and volunteers, if not using it as justification to invade or initiate a border war (similar to that seen in Manchuria) in protection of proles being oppressed by the capitalist Poles.
How a stronger Poland would have affected Germany posturing and expansion would also be interesting. Unfortunately, it is likely the western allies would have maintained their policy of appeasement, which would begin as early as 1926, preferring warmed relations rather than strained ones with Germany, even if there was a stronger counterbalance to them in Eastern Europe in the form of Intermarium / Poland. One thing I can picture changing is the German ultimatum for Memel, in which case, I doubt they would even send it, for they never challenged Poland historically as early as 1938, let alone a strengthened one.
Caught fighting a war in the eastern borderlands, and with large minorities of dubious loyalties in both the west and east, Poland would find itself once more in a precarious position, even with a strengthened military, they would likely succumb to a joint German-Soviet invasion as they had historically, albeit showing hopefully stiffer resistance and lasting longer before capitulating.
edit: I reccommend reading Adam Zamoyski's Poland: A History to better grasp the circumstances