If I owned a gym then I'd message everyone saying that I would like any transgender member to contact me in private to discuss any problems that they or anyone else might be experiencing or they think may experience so that we can work together to address them.
My guess is that signs saying 'Only Biological Women Allowed Here' wouldn't be one of the solutions.
Unfortunately you are now not living in an America that will be cognizant of that view.
There would need to be a more upstream solution, or some sort of "agreed upon" standard.
Just spit balling, I'd suggest that Norway has the right idea with how they approach the situation.
In a nutshell, in order to legally change your gender in Norway, the requirements they have are
- you must be at least 16 years of age
- sign off by a qualified physician stating that the person, indeed, has gender dysphoria and cognitive/behavioral therapy has not worked
- approval by a county medical officer (who reviews the doctor's assessment)
- once the hormone therapy/surgery process has begun, you're assigned a new "Norwegian identification number" (the equivalent to our social security number), which means your old identity (both legally and financially ceases to exist), which means you'd have to reestablish relationships with financial institutions, insurers, etc...
process takes about 2 years and at the end you're assigned a whole new set of documents (just as if you'd been born as that gender)
I learned of the process they have via an interview that was done with one of my favorite comedian's Jim Norton, who was interviewed alongside is wife Nikki (a trans woman from Norway), and the host did a 1-hour deep dive with Nikki asking what the process was in Norway and how that compared to what's going on the US. (interview is way to "blue" to post here, but can be found on youtube)
Nikki's stance was the same as Buck's, which is (paraphrasing) "we're concerned that too many young people are being encouraged to pursue this way too quickly, this isn't (nor should it be viewed as) a costume that one can simply put on and take off on a whim...this is a long journey and process that requires a lot of thought and consideration, the fact that you Americans are letting young kids move forward like this is crazy"
So while I'm sure you still have some of the young people over there doing the whole
"Well, achhctuallly, I'm a non-binary, gender fluid neurodivergent two-spirit, and my identity changes regularly so you have to check in with me once a week to see what my pronouns are", that sort of thing isn't legally recognized legally or for sex-segregated facilities, sporting leagues, etc...
Meaning, their women's sports leagues and changing rooms, legal recognitions, etc... would be restricted to biological women, or those who've gone through the aforementioned process. Not the people who started identifying as trans last month and haven't so much as consulted with a therapist yet.
From what I've read, I'm led to believe that the other Scandinavian countries are similar in that regard.
I'm guessing, in no small part, due to the fact that their healthcare and university systems are nationalized and publicly funded. Meaning they have a vested interest in having some sort of standardized process and don't want to have to micromanage in the way one might under the ruleset of "I can identify as whatever I want (or nothing at all) starting now, and that could change again in 2 weeks"