Very best of luck to you Emma.
A question I'd also be asking, with all due innocence, is where women are supposed to go if they are unable due to belief, culture, disability or trauma to use a mixed sex space?
Or is the gym exclusionary towards those protected characteristics, and only selectively 'inclusive'? In which case this should be transparent and clearly labelled as such.
Obviously women cannot be expected to disclose such very sensitive data to staff - and who on the staff is qualified to pick up the pieces of the issues that such a disclosure to a stranger in a public place may cause - and it must be assumed the gym hasn't thought through how they might be responsible for receiving such sensitive data and storing it because that is a minefield of its own.
As you cannot tell or require women to state whether they require single sex facilities for those protected characteristics, the responsibility (unless you have chosen to openly exclude women of those groups) is to provide this alongside a range of other facilities so that women can select the one that works for them. It's rather like a GRC works.
Why yes, you will have the occasional male who on principle won't use a mixed sex women's space and will want, for his own reasons, to use the female only space to prove that there is no territory that he cannot control. However it is then the responsibility of the gym to police that boundary and if necessary warn individual members that they have the alternative of a sex based space or a mixed sex space, they do not need and cannot reasonably expect to have all the spaces, and by so doing exclude some women. And a member that is unwilling to respect the needs and access of other members can then choose between doing so or having their membership rescinded.
Yes, everyone will be able to tell.