I think I'm going against the consensus in this thread, but I actually don't agree the OP is handling it well, because it sounds like you're being just as simplistic as your daughter is.
It may not be welcome in this forum to say it, but there's a real debate to be had here.
I consider myself a diehard feminist, and I hate seeing how this battle is being played out between women, who have been and are badly discriminated against, and transwomen who are at least as badly (and in many cases probably more) discriminated against. Surely, we have more in common than what divides us?
Some of the language that the OP is using ('brainwashed') suggests that the OP doesn't acknowledge any room for debate and I can understand how the daughter finds this frustrating. I too want 'to live in a world where transpeople are accepted for who they are' and I think this impulse of hers is coming from a kind and positive place.
I won't go more into this or into my own background as somebody who has heavily used women-only spaces of all kinds, but I think there are worse life philosophies than 'be kind' and 'be open and welcoming to people who we perceive as not like ourselves'.
And for @AnnaMagnani (good name!) who said: And no, there is never any evidence that any of them has ever met an actual transwoman in real life. It's all posh women with luxury beliefs
I work in a university, and I've met a number of transwomen, though work or in private life, over a period of twenty years. And I don't see myself or my colleagues in the description "posh women with luxury beliefs"! :)