Their hatred of trans-women seems in large part an extension of their hatred of men.

Exactly, I totally agree with this. It's good to see people like you speaking the truth. I'm GC, I'm female and I'm saying this. Most GCs only ever criticize TIMs, they say "don't invade spaces of women" only because they say "men are stronger than women" but when a TIF invades spaces of men they are like, "meh, good, it doesn't matter because she's not a threat to men and men can defend themselves". Their whole argument rests on their hatred of men, that "women are weak and men are strong, most men are aggressive rapists and they shouldn't be in women's spaces but if a woman goes in men's spaces it's fine because women are sweet and weak and pose no threat to the brute men"

Like, no. Invasion of space is invasion of space. A TIF has no right to invade the spaces of men, just as a TIM has no right to invade the spaces of women. It doesn't matter that "a TIF poses no threat to men", if a woman is a master of martial arts and a TIM that's weaker and smaller than her comes invades her space, do these people say "meh, it's fine, the TIM poses no threat to the woman"? I'm sure they don't