And how do you prove that a declaration of someone's internal feelings of gender identity was falsely made anyway? By definition 'I believe this at this moment in time' can never be proven to be true or false.
This is the absolutely central legal point, to my mind, OldCrone (and one I've been banging on about for years on this board, but you've encapsulated it beautifully succinctly).
Any sort of legal construction (nationality, money, corporations as "persons") requires legally definable and falsifiable criteria to enable you, the judge, the jury, the government, to tell the difference between circumstances where that construction is in play, and circumstances when it is not (and circumstances where it's being invoked fraudulently). Gender self ID relies entirely on what people say about the contents of their own minds - it's unverifiable, unfalsifiable. It's a legal nonsense.
If Karen White says "I feel like a woman" his assertion is no more, nor less, valid than, I dunno, nice Karen in accounts who has just quietly got on with life, turning up to work in a twinset and pearls rather than a suit, for the last 20 years and leading a blameless life. Self ID and "acceptance without exception" do exactly what they say on the tin: once a man has uttered the magic words "I feel like a woman", he is teflon. There is, in principle, no evidence that could be adduced to show that he is in fact a lying predatory bastard.
The current GRA, coupled with the exemptions in the Equalities Act, however imperfect a compromise it is (and I think it is pretty dreadful, and was deliberately "sneaked in under the radar"), does at least attempt some sort of meaningful social contract, namely "we will collectively as a society engage in a polite game of 'let's pretend', so long as you put yourself through some process of gate-keeping where a couple of third parties get to interview you and see if they think you're genuine, and what's more, we reserve a range of circumstances where biological sex trumps the game of 'let's pretend'."