There are two questions that should be required before you even discuss how you think that transgender people should have all rights and privileges enjoyed by cisgender people of the gender they are transitioning to.
For this thread, I'll talk about transwomen who want to be able to use which ever bathroom, changing room, or locker room they choose. Specifically, biological men who identify as women and expect others to affirm that by saying nothing when they go into private women's spaces.
If you support that, your two questions are:
1) How far along in the transition does a bio-male have to be for us to have to accept that they are a woman or girl? Do they have to dress like a woman? All the time, or can they just change into a girl's swimsuit for the meet, and otherwise avoid feminine cothes or makeup? Does it not even matter because "girls clothes" and "boys clothes" are just social constructs anyway?
I'm chubby, bearded, balding, and wear men's clothes. If I were to walk into a women's bathroom or locker room, and there are people around who notice me walking in, should they say anything, either to me or to management? Or should they assume that by walking into the women's room, I indicate that I am a transwoman and mind their own business? If they ask and I tell them I'm trans, should that be the end of the discussion?
If that is not enough, then where do you draw the line?
2) Would you have sex with a pro-op transwoman? Assume that - other than having a penis - she is attractive enough for you to be interested if the opportunity was there. If you're married or otherwise committed, you can still answer the question as a hypothetical. I'm happily married also, but we won't go to heck or have to sleep on the couch if we speculate about whether we'd have sex with the St Pauli Girl vs. the Bud Lite Girl.
These questions are key to your being sincere abut supporting transwomen and girls "rights" to female private spaces. If you would not have sex with a pre-op trans because the penis would make you so uncomfortable that you could not enjoy sex, then you have no business at all ridiculing and mocking a fourteen year old girl who would be uncomfortable in a locker room with a girl or woman with a penis.