This is a tricky and highly emotive area - from what I can tell the organisation in question has the local state laws on their side and there's not much I can do about that. As Sarah notes, they have the "right" to state who they permit to be a member - that's a legal right, even if it is not morally right.
The way I see it, as a female only gym, their principle mission is to offer an environment for their members that is not only safe but also feels safe and unthreatening. Presumably they draw a line on what can be perceived by members as unsafe and threatening based on the law. A member can't say they find a disabled person unsafe or threatening, that would be illegal. A member can't say they find someone of a different race and/or religion as unsafe or threatening, that would be illegal. At this gym, in my home state, a member can say they find a female person with a penis to be unsafe or threatening, and the club has the legal protection to enforce that - presumably by asking "suspect" individuals to prove the status of their genitals.
And I am totally 100% for that gym providing an environment that is, and also feels, safe and unthreatening. What's at issue here is that, as a trans woman I should not be perceived as unsafe or a threat to that environment. And neither should any trans woman, regardless of whether they are pre/non/post op - there are rules for members to follow and one would expect that, cis or trans, they would be expelled from the gym for any transgression. It's not that the gym can't make the gym safe for all women, it's that many cis women feel uncomfortable with trans women. Any hateful prejudices aside, most of that discomfort boils down to not knowing or understanding the trans condition; the vast majority of cis people don't even know that we (usually) take hormones, don't understand that this isn't a "lifestyle" choice, don't believe that it isn't some sexual kink, etc. When you're up against that degree of ignorance it's no wonder organisations come up with stupid rules like "prove you have a vagina". Is it for the organisation's piece of mind, or their clientele's or both?
The crux therefore is how far is the organisation prepared to go to prove that it is an organisation for all women - be they cis or trans or intersex or whatever? If a member was to complain that they were uncomfortable about a disabled person or a person of different ethnicity/religion they would be told "too bad, get over yourself, they have a right to use these facilities and we support them doing so - if you don't like it you can leave". So why not do the same for trans women? It would need to be done in conjunction with clear unambiguous statements in their contracts, staff training, advertising material, gym newsletters and info sessions. Work with trans members so they can demonstrate there is no threat and no need to feel unsafe. Ensure any breach of that safety, from any member cis or trans, results in expulsion. As Squircle noted above, her partner also admitted to feeling intimidated by certain cis women.
Anyway, that's what I'm working towards taking to the gym. They are a national franchise so I don't know about my chances but I have to start somewhere...