I'm not so sure. Most pro-LGBT activists generally held anti-Cold War attitudes that were common in the the Western Left. Meanwhile, most anti-communists tended to be socially conservative and disdainful of homosexuality. In light of that, I have to agree that an LGBT-tolerant USSR would actually be harmful to the gay rights movement in the West. At worst, we might see conservatives advocating Old Testament attitudes towards homosexuality, i.e, stoning.
Some of the most extreme conservatives would no doubt go that far, but this hypothetical situation comes close to IOTL, anyway.
Also, even if what you said about most pro-LGBT activists holding actively anti-Cold War attitudes was accurate(and it may be partly true, at least), it does nothing to disprove my theory, TBH.
As it stands, it's actually quite logically sound- the idea that the United States should try to "out-civilize" the Soviets, in order to win the Cold War, was always fairly prominent in this country, especially from the later '60s and onward. I have no doubts that many more hardline conservatives would no doubt react with horror and disgust if the Soviet Union were to fully recognize civil unions, in say, 1980 or so(gay marriage might be out of the question for a little while longer, though), but it's honestly not that hard to get at least much of the rest of the public on board, gradually(over a period of, say, 10-15 years or so), that is, with the general idea of out-advancing the Russians in general, including socially.
Also, when you think about it, given the particular circumstances, things really did get about as LGBTs as they could have IOTL, by and large, at least as far as the U.S. is concerned; it is rather easier(slight understatement, btw!) to create a plausible scenario in which a *U.S.(that doesn't fall into a Rumsfeldia-style dictatorship and/or suffers an apocalyptic downfall, etc., that is.) starts a legal road to gay marriage some years earlier than it did IOTL(I'll grant you that, without a Supreme Court order, ~1975 is definitely a tad on the optimistic side for the most part.....but 1985-90 shouldn't be *too* hard, especially if you weaken the Religious Right even a tad compared to OTL, and ~1995 is definitely within easy reach), than it is to delay it much, or even *any* further than 2004, when Mass. fully recognized gay marriage IOTL.