This comes up many, many times!
As FWR threads so often say if the word women / woman are used then it is single sex. This has always been the case prior to the Supreme Court ruling under the single sex exemptions within the EA.
The problem was and still is that too many people subscribed to Stonewall training which wrongly told people that if someone "identified" as female they could access women only services.
Many refuges are women only under the SSE, but this doesn't mean they dont also offer services to men, and TW.
The point is, and always should have been that service providers must be clear if their service is women only or trans inclusive.
That was the basis of Roz Adams taking her employers (I cant even remember now which women's groups in Scotland it was) that as the providers of a service that must be honest to those they are providing the service to. So even though she believed in trans inclusive services, she did not accept that she should life to those seeking support.
The reality is the actual law has always been there.
The problem was and still is that too many people, including politicians decided to believe Stonewall and others.
Not just to "be kind" but because as always women's rights are not considered important.
You could always just ask the charity or the council what their policy is.
And see if they answer with mumbo jumbo or are clear about whether it is women only or trans inclusive.