Let me nail my colours to the mast.
Do I think that being transsexual is a "real thing"? Yes - I think there are people who genuinely and sincerely wish and believe that they ought to have been born in a body of the opposite sex.
Do I think it's possible that such feelings have some sort of biological underpinning? Yes, though I think the causes are multifactorial - there's a possible biological component, and a strong social component (as evidenced by the fact that the anthropological evidence suggests that so-called third genders are much more common in societies with very rigid sexually stereotyped roles for men and women).
Do I think that having such feelings means the person in question actually is or can become the opposite sex? Emphatically no.
Do I believe that people who are trans should have human rights and be treated with kindness and respect? Yes, of course.
Does that extend as far as giving transwomen (born biologically male) women's rights? No, because women's rights are there to redress the imbalance in rights we have due to the unique circumstances and vulnerabilities which come from our biology.
For these reasons I'd fully support specialist trans wings in prisons (because to leave transwomen in the male estate does leave them vulnerable in all sorts of ways a civilized society should not countenance), but will never support putting transwomen prisoners into the female estate.
The same arguments extend to women's sports, women's domestic violence shelters, women's rape crisis centres and women's right to ask for a fellow biological woman to conduct intimate examinations if she wishes.
I personally am happy for transwomen to be treated according to the legal fiction in pretty much all other spheres - if a transwoman Jane wants to go to work as an accountant and be called Jane and referred to as she/her by her colleagues, and not subject to harrassment or discrimination in the workplace, that should of course be her right. I will engage in that polite legal fiction. I will not extend it to circumstances where women will be harmed by that polite legal fiction.