My gut instinct is to say that trans women are women, period, and therefore should be permitted to do anything any other woman can do. There is however, a logical dilemma at the heart of this debate that seems to inevitably and necessarily make it impossible for trans women to compete as women. Here is the dilemma in logical form:

transwomen are biologically male, even though their gender is female.

If biological males have an advantage over biological females, they should not be allowed to compete in women’s sports.

If biological males do not have an advantage over biological females, there is no reason to have a separate category of women’s sports.

Either biological males have an advantage over biological females, or they do not.

Therefore, either transwomen should not be allowed to compete in women’s sports, or there should be no separate category of women sports.

Regardless of which horn of this dilemma you grasp, transwomen don’t get to compete against ciswomen. Either they are excluded from women's sports, or there are no women's sports for them to be excluded from.

The logical structure of this argument is valid, so the only way I can avoid accepting this conclusion is to find some way of disproving at least one of the premises. I would welcome this, but I can't see any way of doing it.