My reply to the ‘I don’t want men putting on a skirt so they can spy on my daughter/wife’ is typically to explain that by pushing back on this rule, what they’re saying is that they’d prefer to have this person, a transgender male, in the bathroom with their daughter, right? Also, they don’t seem to understand, not that I’d expect them to, that a man throwing on a dress isn’t trans or gender fluid, he’s a cross dresser and even would still have to use the mens bathroom. Wearing women’s clothing and wanting to be a woman are two different things.
Of course, the few times I used that argument with the unwashed masses, many of them went straight to ‘I’ll shoot anyone that looks like that if they try to use the bathroom while my wife/daughter is in there’.
That’s the problem, they (whoever ‘they’ are) don’t want a man “pretending” to be a woman to be allowed in the women’s room. And as much as they play the ‘if you’re born m/f, you’re m/f, period’ game, they also don’t want someone that’s born a woman and now presents as a man, to be in the women’s room.
The general take away from that is, so far as they’re concerned, if you don’t look and act like the gender you were born as, you just need to stay home.
ETA, that picture is Buck Angel. There’s a good documentary on him out there somewhere, but he’s a porn start so it’s very NSFW.