Even the alphabet soup is somewhat iffy since there's already some stuff going on there. House Amaris was started by a lesbian, for example.
As for trannies, can't kill what's not there. The Clans' eugenics programs only apply to the Warrior caste (and a couple other Clan-specific minor cases), and it would heavily select against sexual deviancy because despite her accepting attitude towards casual sex they're still pretty conservative otherwise. Add to it that gender identity isn't important for canister-born Warriors raised in coed environments since birth, and there's just not much space for the social infection to spread. Plus, if they go "IT'S MA'AM!" so hard it triggers a Trial of Grievance, eventually they'll challenge the wrong "transphobe" and get killed. Or irritate their superior and get set to investigate these woods over there for Steiner Scouting Parties in an UrbanMech IIC.
Trannies in their civilian castes would also get no support because, again, conservative society and troons' "needs" for ill-fitting clothes and superfluous medical procedures to feed their vanity would not be looked upon kindly. And the Clans are an authoritarian police state. Anyone trying to start their own little gendercult in the Chatterweb would get a housecall by some Elemental Watch officers really quick.
As for the Inner Sphere, while galactic-scale politics must be conservative due to arranged political marriages being not only common but defined as being between a man and a woman, individual planets could have their own things going on. Barring specific events in their own history, though, after three Succession Wars a lot of planets in the Inner Sphere would simply not be at the right level of economical prosperity and population development to even give a shit about First World Problems like gender politics. Sure, Tharkad or Atreus with their populations in the multiple billions might have a "vibrant" LGBT community depending on who's in charge at the time, but a 6-million inhabitant rock in the middle of the Draconis Combine like Port Arthur? Good fucking luck.