FunctionalAnatomy, that's interesting that you were able to persuade one friend almost instantly, and the philosophically minded one not at all. I wonder if the latter's education featured a lot of postmodern philosophers? It seems that people steeped in postmodernism are the most vulnerable to magical thinking about biological sex not existing.
This is going to be a bit long, but I've had many arguments about gender identity in real life and online, and I've thought a lot about the best way to cut through the propaganda and make people think. I've convinced my male partner, a gay male friend, two female friends and my mother, all of whom were previously more or less pro-trans, with variations of the following arguments. (Note: All of those conversations took place face to face - I think it would have gone differently if I'd tried to argue with them on Facebook about it, due to the performative nature of social media.)
First of all, I think the most crucial thing is not to argue on the terms set by trans activists, and, especially, to speak plainly rather than using the Orwellian language they demand, which is deliberately designed to linguistically erase biological sex and obscure the power relations between men and women.
So don't say 'trans women', say 'trans-identified males'. And don't frame the issue as being about the competing rights of 'women and trans women'. Most people will always agree that the former should cede to the latter, especially because to them a 'trans woman' is an old-school transsexual who has got rid of his penis, so what are us silly bitches even worried about, eh?
Talk instead about the abolition of sex segregation in prisons, hospital wards, women's shelters, changing rooms and sports (as well as the abolition of reliable sex-based crime statistics), because that is what enshrining 'gender identity' into law actually does. Talk about how many men who are now identifying as 'trans women' have no significant sex dysphoria and are what used to be known as fetishistic cross-dressers. Point out that under self ID of gender, 'trans women' are legally and practically indistinguishable from men as a group. This isn't about letting a well-defined, numerically tiny group of males with sex dysphoria into women's spaces. It is actually about the abolition of sex as a meaningful political and legal category, as well as enshrining a deeply regressive understanding of gender into law.
Don't get into debates about the supposedly sky-high rates of trans suicide and murder (although do take the time to point out that trans-identified males commit violent and sexual crimes at the same rate as other males). Talk instead about how 'gender identity' itself is a sexist and irrational belief system that has no objective qualities. Ask them what it means to claim that a male can 'feel' female and vice versa. How, exactly, can a man 'feel like' a woman? What characterises this 'woman brain' or 'woman soul' that can supposedly exist in male bodies? Point out that children are being sterilised and given dangerous drugs in the service of the insanely sexist and illogical notion that a male child can be born with a 'female soul' inside him and vice versa. You may not get through to everyone instantly, but just by raising these questions you will have planted seeds in people's minds that maybe the trans issue isn't as straightforward as they thought.
I also think that short, clear statements are best for cutting through:
Eg, simply stating: 'A woman is an adult human female, not a feeling in a man's head', is an effective counter to the 'trans women are women' catechism.
Even just stating 'transgender ideology harms women' out loud helps to make people think. Maybe some women who had been previously unthinkingly supportive of trans rights, while they may not agree with you right away, will have had the notion that it is permissible to question this planted in their heads, and when they see something that is not right (say, a story about a male murderer wanting to go to a women's prison), they will have a frame of reference for 'harm to women' in their heads that they can slot this into.
Instead of desperately defending ourselves against charges that we are 'bigots' for rejecting gender identity doctrine, feminists are strongest when we calmly challenge the illogical and sexist nature of the doctrine itself, and when we point out the real-world harmful effects that it is having.