This depends a lot on how badly things degenerate before the revolution s defeated. Also, dealing with internal unrest will probably be easier once the war is over, especially if the regime can reap some of the benefits of being a theoretical "winner" in the war. Also, a white Russia would not have to deal with foreign intervention and would instead enjoy the support of it's former allies, at least in the immediate postwar years.
Anyway, let's assume for the sake of argument that the Russian empire survives in a good enough shape to be able to throw it's weight around on the international stage immediately after the war ends. How does this affect international relationships?
Russia post civil war will not be in good enough shape to throw its weight around. If you are looking for a realistic scenario, that's not it. If you want fantasy scenario with no bearing in reality, go ahead, but realize you are writing complete fiction with little based in reality.
I think you are being way too blase about the true nature of Russia's internal politics. Things had already badly degenerated before the February Revolution had even happened. By the October Revolution is was even worse. The actual civil war plunged everything into chaos and horror. The time when anyone could put back the Russian state without terror and bloodshed is over. All the diseased pus of the Tsarist state was coming out.
The winner of any Russian Civil War won't be getting any benefits from being the winner except that they've eliminated their main competitors and thus wield centralized power enforced by the gun and bayonet (a big direct benefit, but not the intangible one you seem to imply). Even the Soviets had to deal with Makhno's peasant partisans, labor unrest, sporadic revolts, and nationalist rebellions. Simply because the Whites are winners doesn't mean they don't face resistance. They probably face even more resistance than the Soviets because unlike them, the Whites never condescended to even pretend in land reform or pro-peasant policies. Internal unrest will be severe for a very long time.
There was a very widespread peasant's movement in Russia before the World War I, during it, and after it. That won't go away. Stalin's eventual solution was collectivization and starving the Ukraine into submission. What are the Whites going to do? This will be the most important decision the Whites must make for their state to survive. It will take years to play out.
What support do you think the Allied are going to give the White government? Neither the British nor French are going to waste time, money, or blood at reassembling the Tsarist empire. They may not be hostile to it, but they won't be interested in helping the Whites suppress the Finns, Poles, or Georgians. Even the nationalist assemblies that petered out during the Red's victories will be around in a White victory, and the Allies will play footsie with them for a while until they determine if they are viable or not. The western Ukrainians might still drag Poland into a war with a White Russia. The Kuban would expect strong autonomy, if not outright independence, after a White Victory. Any White government is going to be obsessed about the aftermath of Brest-Litovsk and all of the subsequent declarations of independence afterwards.
Much of Russian "foreign policy" in the next ten years will be based on dealing with these de facto independent nations. You simply can't handwave that away.
You really need to define how the White state developed - is it a weak pseudo-democracy like much of Central/Eastern Europe in the interwar years? Does it lean to a liberal democracy? Is it autocratic? How does it treat the ethnic minorities within its power? Which lands did it successfully retake? Which ones slipped from its grasp? Only then will you know what the Russian foreign policy objectives are.