White, middle aged men telling us to be kind...

Yeah of course he says he's comfortable with it whilst foisting the problem onto women, sounds like a big fat liar to me.
Maybe he should work on changing the mindset of the men that make them uncomfortable rather than telling women to budge up. They're not ours and we don't want them.

I have a friend, (a father to daughters!), who also doesn’t seem to have thought too hard about the inherent issues in parroting the TWAW mantra.

He’s generally a pretty intelligent guy but I think he’s had one too many EDI / two-minutes-hate sessions at work, mixed with having a friend whose teenage daughter has decided she’s trans (and has been bullying her folks into obtaining PBs for her online, on threat of suicide if they won’t).

He was “very disappointed” in me when he realised I strongly disagreed with him. I didn’t care.

Rant away! It's infuriating.
Men, as well as being bigger, stronger and faster than women, commit over 90% of violent crime and 98% of sexual crime. That doesn't mean that I think that all men are dreadful and dangerous (I like men!) but at population level there is a social issue that has to be dealt with, and has to date been dealt with by sex-based rules and rights.
The social issue doesn't go away just because of a group of men being, or claiming to be, trans.
People like your BIL need to engage their brains.

I have a DSS like this. luckily my DP is very much on the gender critical end of the spectrum, so I haven't had to divorce him. Due to having a difficult relationship with DSS in the past and some serious mental health issues he had, we just cannot risk upsetting this particular applecart. But this is a white, straight 33 year old man, with no wife or daughters, and very little in the way of empathy. This issue is just a no risk way (to himself) of signalling what a liberal right on chap he is, without having anything affect him personally or having to think too deeply. It fair boils my piss.

Try the Staniland Question on him. Ask what his foolproof method for separating out the wrong 'uns from the nice ones before a girl is harassed or harmed is. When he admits he hasn't got one, ask for a ballpark figure of how many girls harmed is unacceptable collateral damage in the interests of not making men sad.

The ones who haven't already blocked me usually shut up at about this point.

Pay no attention to him, he sounds like a complete and utter wet wipe. Cannon fodder for when there is an uprising.

Exactly, tell him men need to be kind by expanding what it means to be masculine, by embracing gender nonconforming men in their spaces and to stop trying to make this women's problem to solve.

Why should we women accommodate everyone if these men don't lead by example?

Oh god a relative? Commiserations (but also thank Goddess he's not your partner!)

I'd be tempted to take this one of 2 ways. Either come out as NB in due course, with a very straight face. Or, as his belief that one can identify out of one's sex class is misogynist, I'd refuse to spend a single scrap of unpaid female labour on anything that benefits him. I'd also consider applying postmodern thinking around other nouns if and when it suits you.