I think there is merit in the OP’s idea.
Bear in mind that Churchill seriously considered war with the Soviets immediately following the defeat of Germany. Anybody heard of Operation Unthinkable?
Once hostilities had broken out with the West, I think it's extremely unlikely that The Allies could have joined forces with Germany under any circumstances. However, Stalin's early moves & behaviour could certainly have had enormous effect on Allied strategy regarding, for example, aid sent to Russia. I'm actually surprised that Britain helped Russia at all considering Stalin's actions of 17th September 39. The German invasion of Poland was the final straw for The Allies. But the Russian invasion triggered no response, after which Britain sent large amounts of aid to Russia following the German DoW.
The USA, of course, sent staggering amounts of aid to Russia. I would state that without this aid from her Western "allies", the war with Germany would have been very different. How much further Germany would have penetrated is up for debate, but that Germany would have penetrated more deeply is, I think, simply obvious. Also, without this aid from the West, Russia would not have been in as strong a position militarily (or geographically) after Germany was defeated. And Germany would have been defeated. Her industry would still have been torn down by bombing and the resources sent to Russia could simply have been used in Italy & France instead.
The concept of MAD during the game period is not, IMHO, appropriate as the early atomic weapons were neither of sufficient power nor available in sufficient quantity. By August 1946, there were only nine cores for the Fat Man weapons complete.
By December 1948, there were 53 Fat Man cores available and there was also enough bomb-grade material for an additional ten Little Boy weapons. Russia conducted her first successful atomic bomb test in August 1949. Not assured destruction, and certainly not mutual.
So… theoretically at least, war with the Soviets should certainly be possible from a number of different scenarios.