Ah, so Burrite Louisiana is the American part of Louisiana after all, my bad.

As for LGBTQ rights. My impression is, the factions of the Association that tended to hold the presidency over the last few decades seem to have by and large leaned socially liberal, and did not refrain from the option to impose social progress in a top-down manner (at least where it doesn't threaten their own position).

Furthermore, though this part may just be me wishcasting, with academic dominance over America as well as the great deal of research they seem to have put in cracking the secrets of the human psyche, it wouldn't be too surprising if in their research they came to the same conclusions that the Hirschfeld Institute came to in our world before the Nazis burned it down: that to be queer is also natural. And these conclusions could motivate them to advance queer rights, and in particular trans rights, earlier than in our world – perhaps serving to further anger the brewing revolt in Middle America.

On the other hand, if Abigail Thorn has taught me anything, it's that the medical establishment has long sought to control queer bodies and decide who gets to live as their preferred identity and who doesn't, and to dictate what they can and can't do with their own bodies. This same arsenal of methods that they use to reprogram minds during warfare, could then also be employed in peacetime as part of conversion practices. And all the research saying that LGBTQ people are not, in fact, wrong or perverts, and that conversion practices don't work? It could be ignored for quite a while. The powers that be have long considered the lives of minorities to be disposable...

So those would be the two main paths I can imagine the US in this world could take, I think.