While I'm in agreement with the argument that the rightward drift of the GOP doesn't have much to do with the fall of the Soviet Union, I'm going to try not to be so dismissive of the challenge. The challenge calls for conservatives to become increasingly moderate after the fall of the Soviet Union; so let's think of ways the collapse of the USSR might proactively influence the situation.
In many ways, there wasn't much keeping the Soviets from embracing some very conservative ideas. They embraced several IOTL already. Let's have them do it louder and more completely, with harsher punishments for rule-breakers.
By the 1980s conservatives in the US are stalled, unable to move rightward socially without pushing into "Pinko Territory." Meanwhile, the aged rulers of the Union are losing control and increase their repression campaigns against gays, women, minorities, etc. to a fever pitch. It gets kinda Cambodian for a little while, with defections going through the roof and the collapse of the Iron Curtain a much more chaotic affair.
But however it ends, it is clear to the world that what won, what caused the communists to overthrow their system, was the desire for Western, progressive values. The victory narrative forces conservatives to either embrace a (limited) acceptance of a progressive consensus, or to reject the evil of the Soviet system. I don't think they'd be willing to do this in the 1990s.
You'd need further events to maintain the trajectory for more than a decade or two, but that's true of most political trends.
