It sounds like you and your child have found a way to work through this together, which is fantastic.
He was using the men's before but can now see it's not ok.
This is interesting and shows the importance of different viewpoints coming together to figure this stuff out at a societal level. The law gives clarity on single-sex spaces which is a useful starting point. I do feel sympathy for transmen, given the "limbo" that some will feel in situations where they "pass" well enough that the law specifically allows for their exclusion from women's spaces too. Yes, they made their choice to look as much like a male as possible but that doesn't mean it'll be easy to live with the consequences of those choices. Obviously there are some transwomen who "pass" too. The third spaces discussion has been suppressed by TRAs but hopefully some of the more reasonable trans voices will start to break through, just as we're seeing on this thread.
I see it a bit like going through the five stages of grief, I've been through them all but have got to a reasonable acceptance, more for my own peace of mind than anything. I can't keep stressing about this forever.
Personally, I would not be able to use opposite-sex (or non-binary) pronouns for my daughter if she ever decided she wanted to transition as an adult.
When I started this my journey to understand gender identity and support my (then actively) gender-questioning daughter, I felt open to the idea that I may need to do so in the future. That I might need to reframe my thinking and accept "him" as my "son". At that time I was prepared to do this if it was the right thing for her and I focused my time on researching what the whole thing was all about - including here on MN.
I do still accept that as an adult she may transition - and yes, I would feel grief (hopefully through to acceptance) because of the significant health risks she would face if she took any medicalised steps. However, I am no longer prepared to alter my own reality if my daughter chooses a path which differs from it. She will always my daughter - in law, biology and objective fact - and I'm not going to say otherwise.
If I shared her belief that gender identity exists (I used to believe this) I would accommodate her belief by using the "correct" pronouns which relate to it - but I don't. However, I would like to hope that even in that scenario I would never have accommodated her expecting others to bend their own reality, either as an overt demand or through emotional blackmail. I'll never know either way because during my learning journey I realised I didn't believe we all have a gender identity. Even prior to this discovery about myself I would never have supported anyone enforcing their belief on someone else. There are many people who hold different beliefs who find a way to co-exist within society.
If this hypothetical situation becomes reality in the future, my concession would be that I'd a) refer to her as "my child" or "my daughter who identifies as a man", without judgement or affirmation b) stop using pronouns to describe her altogether - no she, they or he. I'd go through the mental gymnastics of complete pronoun removal when describing her (I appreciate this isn't easy - as per Rapid's comments - and I may find I can't do it because my brain gets muddled) c) I would accept other people using my child's preferred pronouns if that's what they wanted to do.
If those accommodations weren't good enough - there's a poster on page 6/7 of this thread saying that the only way to be a good parent is to actively use preferred pronouns - I'd reluctantly find a way to grieve and accept my child's decision to pull away from me. Right now though, I'm fighting as hard as I can to avoid such a scenario because the idea of it breaks my heart and keeps me awake at night. But what I won't do is bend my reality just to avoid it.
We all have different lines on this as parents and children. No one family's way is better or worse than another's if they can maintain a way to communicate with each other when there are differences of opinion on an (adult) child's transition.