On the whole the freaking out about "trans" stuff is mostly a red herring or strawman used for strategic propagandistic purposes more than it is reflecting any real world problematic.
Making outlandish claims about how liberals are trying to do X, Y, and Z to their children's education or sports forces liberals onto the defensive forcing them to defend hormone therapy for 12 year olds or whatever when the actual number of real world incidents of this is vanishingly small.
I don't personally carry any kind of radical gender identity theory, but I care about the rights of minorities and it's none of my business what other people do and I see no problem with the public and corporate sphere accomodate pronoun usages or even bathroom accomodations, it's unlikely to cause harms and it doesn't hurt me in the slightest.
It's also somewhat ... interesting ... to bring up defense of female prisoner's rights and their conditions as some kind of demarcation point.
Is the socially conservative right concerned about humane prisons now?
Because that would be great. But unlikely.