I did remember - maybe misremembering - from the SCJ and the original unfucked about with guidance, that the onus was on the person being difficult to prove their sex, and if this was becoming a battle/unreasonable, then the answer was to deny them access to the single sex and direct them to the gender neutral.
Legally - They're obviously being bloody difficult for their own reasons, it's an unreasonable behaviour that threatens other people's inclusion and access as well as sucking up time and energy from service providers, and you cannot sacrifice the rights and needs of every woman in a single sex space for someone who feels like being bolshy. They still have access to a space.
The idea that they would win would have to be based on the needs of all the women in single sex provision being nothing when weighed against theirs. Which is what is going on in the head of an activist, they're sexist to the core, but isn't going to wash in court.