In my country (spain) a Trans bill was recently passed. Many people has complained about that. The most vocals were cis heterosexual males. Seconded by cis women.

Of course transpeople need equal rights, including bathrooms and gyms. In fact many complains remind me a lot to the ones we got about lesbians in the locker rooms (it was all bullshit). Fast forward to now. Nothing happened, lesbians didn't harassed other women and now is perfectly normal. I think (hope) the same would happen with transpeople.

The point is that conservatives always try to shift the blame for their bad behaviour so we don't focus on all the ways they are being asholes. For example, when they claim that muslim would come and enforce their sexism on women to cover for their own sexism. Or claim that women act like sluts to not address rapeculture.

I think that their hate of trans people is the reflection on their own sexism. It basically say that all their claims about "women being naturally nurturing" and therefore expected to do all the caring, emotional labor, house chores, etc is bullshit. Or that men aren't "naturally better at leading/paid work". It threatens the free labor and extra power they are getting.

You are absolutely right to say that they are grasping at straws and just looking for excuses to deny human rights to part of the population. And if they get that less favoured groups fight among themselves, the better for them. Just hope people would realize that.

Best wishes, I really hope that you can live your life to the fullest as yourself.