We’re missing the point here. The OP mentioned this:
“…since it's becoming increasingly accepted that gender is less about chromosomes, appearance or reproductive ability but more a matter of state of mind…”
Which I think is becoming more evidently true every day. That is all well and good for those who wish to “identify with” a different gender to the one which their physical attributes suggest they should do. But it is not much fun in some specific situations, for those around them who have decided to stick with what nature gave them.
“And my point remains the same...if a trans woman is really some creep bloke who wants to spy on women, or assault them - in a female changing room, or toilets, he will stick out like a sore thumb, and will be easily identifiable.”
And what is actually to be done about a person who is identified as such? He will have been told that his gender is a fluid concept and (effectively) he can choose whichever he likes – “just sign here, guv!”.
“…so how would you tell the genuine from the creep?”
You can’t. And that’s why the creep should not be allowed access to the female locker room simply by signing a bit of paper. And why should he have to dress up like Lily Savage anyway? I don’t see anything that suggests a self-certified female (who is actually a male) has to dress like a woman. All he has to say is “I identify as a female”. Along comes plod to arrest him and he simply produces his self-certificate.
You keep on going on about gender dysmorphia, sp. And I keep going on about people who have simply signed a piece of paper saying they are no longer a man but a woman. You seemed to have accepted the difference a while ago but now seem to have retracted.