That is kind of challenge, as it is not one of these "it almost happened IOTL" scenarious, but still propability is bigger than 0% and time travel or UFO landing is not required for IT to happen. Of course, Romanov PLC is easily doable with late 18th POD (for example Konstantin Pavlovich enforced on the throne) but I'm thinking about earlier times and mid 17th century POD. What is required? Change of personality of Tsar Alexei I and some unfortunate infants living to adulthood.
So the first step is: make Alexei less religious and more supportive for ideas of Ordin-Nashchokin. Afanasy Ordin-Nashchokin regarded Swedes and Ottomans as worse enemies than Poles and for him access to the Baltic was more valuable than unstable Ukraine ruined by decades of war. He also advocated for more moderate peace terms with PLC in Andrusovo, while according to Tsar no Orthodox land should be left for Poles.
Second change: these infants, who would be more fortunate ITTL are Dmitri Alexeevich, oldest son of Alexei (IOTL 1648-1649), and Maria Anna Vasa (1650-1651) daughter of John Casimir.
IOTL Polish delegation at Niemieża, desperate to achieve peace in the East in face of Swedish invasion, promised election to Polish throne to Alexei-that was at the time when PLC was in deep, deep trouble, otherwise such radical proposition would never be made and Orthodox Russian Tsar as King of Poland was hardly something that would be accepted. ITTL at Niemieża suggestion of marriage between Russian tsarevich and Polish królewna would likely appear for the first time and may not be forgotten even if Deluge is over. Polish Vasas were not willing to give up their claims to Swedish throne even when their chances to return to Stockholm were practically non-existing. Thus John Casimir, who has living daughter ITTL, may hope, that alliance with Russia could make it possible for his descendants to return to the throne that is "rightfully" his. Romanov candidature should also get support of Lithuanian nobility (as Grand Duchy of Lithuania, being weaker side in conflicts with Russia, was most interested in keeping peace with eastern neighbour).
So we have these changes:
-John Casimir has surviving daughter, whom he would marry to Alexei Alexeevich (who ITTL has living older brother and is not his father's heir in Russia). So Conde and d'Enghien are not even considered by him as possible successors. There is no Vivente Rege campaign (IOTL Louis XIV paid for it, unlike L14 Alexei has no money for it). So likely no civil war in PLC. After Deluge there is little love for foreigners, but Alexei Alexeevich at least would be familiar with language and customs-Polish language at the time was commonly used at Russian court. And unlike French dukes Tsarevich offers significant political gains. And if said Tsarevich is married to heiress of Jagiellon blood he'd have obvious advantage over other candidates.
-PLC-Russian war after Deluge: if Tsar wants his son to be elected he'd be more willing to make earlier peace with PLC, that is true also for John Casimir-IOTL he hoped, that success in the war with Russia would increase his chances to enforce Vivente Rege, ITTL he would not have such motivation.
-Habsburgs and Ottomans: Ottomans obviously would not be happy (at least Russia and PLC would have common enemy for the next generation), but what about Habsburgs? PLC and Russia conflicted with Ottomans would be Habsburg allies, but if your allies are too strong, it also is not good.
So the first step is: make Alexei less religious and more supportive for ideas of Ordin-Nashchokin. Afanasy Ordin-Nashchokin regarded Swedes and Ottomans as worse enemies than Poles and for him access to the Baltic was more valuable than unstable Ukraine ruined by decades of war. He also advocated for more moderate peace terms with PLC in Andrusovo, while according to Tsar no Orthodox land should be left for Poles.
Second change: these infants, who would be more fortunate ITTL are Dmitri Alexeevich, oldest son of Alexei (IOTL 1648-1649), and Maria Anna Vasa (1650-1651) daughter of John Casimir.
IOTL Polish delegation at Niemieża, desperate to achieve peace in the East in face of Swedish invasion, promised election to Polish throne to Alexei-that was at the time when PLC was in deep, deep trouble, otherwise such radical proposition would never be made and Orthodox Russian Tsar as King of Poland was hardly something that would be accepted. ITTL at Niemieża suggestion of marriage between Russian tsarevich and Polish królewna would likely appear for the first time and may not be forgotten even if Deluge is over. Polish Vasas were not willing to give up their claims to Swedish throne even when their chances to return to Stockholm were practically non-existing. Thus John Casimir, who has living daughter ITTL, may hope, that alliance with Russia could make it possible for his descendants to return to the throne that is "rightfully" his. Romanov candidature should also get support of Lithuanian nobility (as Grand Duchy of Lithuania, being weaker side in conflicts with Russia, was most interested in keeping peace with eastern neighbour).
So we have these changes:
-John Casimir has surviving daughter, whom he would marry to Alexei Alexeevich (who ITTL has living older brother and is not his father's heir in Russia). So Conde and d'Enghien are not even considered by him as possible successors. There is no Vivente Rege campaign (IOTL Louis XIV paid for it, unlike L14 Alexei has no money for it). So likely no civil war in PLC. After Deluge there is little love for foreigners, but Alexei Alexeevich at least would be familiar with language and customs-Polish language at the time was commonly used at Russian court. And unlike French dukes Tsarevich offers significant political gains. And if said Tsarevich is married to heiress of Jagiellon blood he'd have obvious advantage over other candidates.
-PLC-Russian war after Deluge: if Tsar wants his son to be elected he'd be more willing to make earlier peace with PLC, that is true also for John Casimir-IOTL he hoped, that success in the war with Russia would increase his chances to enforce Vivente Rege, ITTL he would not have such motivation.
-Habsburgs and Ottomans: Ottomans obviously would not be happy (at least Russia and PLC would have common enemy for the next generation), but what about Habsburgs? PLC and Russia conflicted with Ottomans would be Habsburg allies, but if your allies are too strong, it also is not good.