Verysadpants, your child is a gender-non-conforming person of female sex, so of course the abuse is worse than what you, a woman who probably performs femininity to some degree, get.
I daresay a young butch lesbian would get the same abuse. Perhaps more.
Does the politics of it have to come at the expense of those who may desperately need their service?
Are you aware that this portraying abortion as something that has nothing to do with a person's sex could mean that the anti-abortion laws cannot be abolished on the grounds of them being sexist, and will therefore stay in place?
Is that really what you want? Do you want millions of women to suffer so that about 1% of the population don't have to cope with a bit of discomfort caused by a symbol and some words?
Your child is underage. If an abortion was needed, then surely, you could access the charity and do all the paperwork, if your child really cannot cope with the cognitive dissonance of accessing a service that's for women.
However, if your child reports sex dysphoria (gender dysphoria, i.e. being unhappy with being assigned the gender role of subhuman fucktoilet, is a very healthy reaction every girl shows to some degree, so if you claim that your child is unhappy with the actual female body, not the male reactions to it, that's sex dysphoria in my book), then I think having the actual abortion would be much, much worse than seeing the symbol and/or word for "woman" somewhere on the paperwork.
I really think there is a big question over whether some transboys would actually survive a pregnancy.
Many girls and women do not survive pregnancy. That is why banning abortions is murder. Misogynist murder. And we have to name it as what it is.
Claiming that abortion is not a women's issue makes this hate against women invisible.