One of my (implied) positions in this debate is that this particular question is in theory partly a question about science.
Medicine claims to be a scientific domain. Mental health claims to be a scientific domain. Obviously our value set will guide our choice of trade-offs. But if medicine were to make a convincing (i.e., scientific) argument that transwomen who are allowed to participate in women's sports have better mental health outcomes than those who don't, that would certainly influence my view of the value trade-offs when making policy.
The main issue here, as I see it, is that the trans-activists want to have their cake and eat it too. They want us to accept, a priori, that transsexuals need certain privileges and accommodations as a kind of treatment for what ails them. But they do not actually produce medical science arguing for these things as a prescription for that ailment.
There is some dancing that happens around the central questions. Obviously medical transition does get prescribed in some cases - cases where the patient actually goes to a doctor for diagnosis and treatment. But trans-activists want to extend the definition of "patient" to include anyone who self-IDs, and expand the definition of "treatment" to whatever anyone who self-IDs claims they're entitled to.