I don't think there's any such list. Battletech in general isn't very LGBTQ friendly because bloodlines and lines of descent are very important to the setting, the Star League literally fell apart because there were no more members of the Cameron bloodline left. The Taurian Concordate schismed and the Calderon Protectorate formed because Grover Shraplen wouldn't recognize an illegitimate son who was descended from the Calderon family and thus continued the bloodline. Basically, anything that isn't strongly pro-family and pro-genetic-kids goes against one of the most foundational themes of the setting.
Assuming you made one...
Probably the Canopians on top given being "space Vegas but with even moar Blackjack and Hookers, plus all Femdom all the time" is their thing. I doubt they'd like men turning into women though, it might be seen as a power grab.
All the clans except Blood Spirit would come next, not only do they strongly encourage cheap, meaningless sex with all genders, they encourage it heavily within the same Sibko (Sibling Company) so there's a pretty strong incest focus*. Pretty much every clan except Blood Spirit are also furries. It's also worth noting that without their iron wombs, I expect the clans would have a tremendous hate for LGBTQ for messing with their bloodline breeding experiments.
Blood Spirit breaks with that tradition and expects kids to be raised by their parents, even trueborns, and their military uniform looks like they're cosplaying as M. Bison rather than wearing fursuits. They get a low deviancy score in comparison.
Pretty much nobody else has much of their sex life discussed outside of "X got married and had kids." Wolf's Dragoons should get an honorable mention for creating a clone of Jamie Wolf except genderbent to create Maeve Wolf so that could be construed as trans support if you tilt your head sideways and blink repeatedly.
At the bottom of the list, Given their cultures, I kinda feel like the Combine and Taurians are probably the closest to Superstraight. The Combine I could see tolerating somebody gay as long as they're in the closet, but said closeted gay person better get with the program and marry and have kids who will fight for the glory of the Dragon in the open. If they don't stay in the closet? The nail that sticks up gets hammered.
Taurians, well they're the American West + Paranoia. While Cowboys are a gay stereotype, my particular observation has been that actual farmers and ranchers are almost never gay, gay cowboys are city slickers cosplaying as the real deal. Additionally, due to their distrust of any outsiders, I think the Taurians would react to, say, a modern-style pride parade poorly. If they tolerated such a thing at all they'd likely suspect it was an attempt by perfidious Davion to destroy their wholesome Taurian culture.
*In the wishy-washy "We're step-siblings so it's not blood incest" porn sense except some of the Sibko could well be blood-related too and there's no indication the clans give a fig.