What will a post-Trump Republican Party look like?

The honest answer is that I have no way of knowing for sure. However, I do think there are some fairly safe predictions, based in part on the data and briefing paper that V-Dem Institute published recently. As noted in that post, the policy positions of the Republican Party had shifted dramatically in an authoritarian direction over the course of this century, arguably accelerated by the Trump era. If the various polling models are correct, and Trump loses his re-election bid (which is highly probable, but not a foregone conclusion, as even the various models acknowledge), will the party begin a dramatic shift toward moderation? 

That seems improbable. Keep in mind that the composition of the party's leadership in Congress is not going to change dramatically, save for perhaps losing a few more remaining members who were at least nominally moderate, and the addition of perhaps some new members who are tied to extremist movements and conspiracy theories (such as QAnon). The RNC itself is probably not going to change much either. So I don't expect the sort of post-election "autopsy" and self-reflection that occurred in the aftermath of 2012. The shadow of Trump will still loom large, as will his popularity among his base of followers, who will continue to follow his every tweet. Trump will not leave office quietly, and post-Presidential life will for a while leave him as a kingmaker. But even if Trump were to go silent, those in power and their voters are not ones reputed for moderation. I would expect a trend towards authoritarianism to continue, even if concerted efforts are made to remove any vestiges of the Trump name. 

It would be interesting to use existing measures of authoritarianism and social dominance orientation developed by political psychologists and fellow travelers to assess the psychology of those in office and among the party's base. Absent that, I can only speculate. I don't expect any change, except perhaps to double down for the foreseeable future. I could be wrong, and would actually be relieved if that were the case.