I know this is none of my business, but before people start demonising this therapist, I would like to ask: with regard to the letter, did you make it clear at the start what kind of letter would be involved? Like what the tone of the letter would need to be, for example?
As you stated, she said herself she isn't an expert in the field of gender identity, so maybe she's never had to write one before. I don't think it is BS to make the point of not being specialised in gender identity, when liaising with someone dealing with such issues. I am currently thinking of training to become a therapist myself, and in all the courses I've looked at, gender issues are a distinctly separate, and involved area within the broader curriculum. It has distinct fields, such as bereavement, gender identity, sexual etc. It's not a catch-all field of expertise. To not have specialised in gender and gender identity is, I think, a valid point to make. Otherwise it would be like expecting your local GP to be able to perform neurosurgery because "hey, they're a doctor, right?"
I understand why you feel the way you do, and you have every right to feel angry and hurt because you feel that someone you trusted has let you down and like you've wasted a huge amount of time and money. But I would also ask: is there any possibility that it was a simple breakdown in communication between the two of you at the outset, and a failure to establish, and agree upon, a mutually satisfactory outcome before taking up her services, rather than any deliberately malicious act of betrayal?
Maybe she is just being difficult and a jobsworth, hiding behind the lack of expertise. That's a distinct possibility. But also maybe there's a possibility that your hopes and overwhelming need to do this led to a greater expectation of what could be achieved with her than was actually the case.
I really hope you manage to obtain the letter you need from elsewhere and get that which you've sought for so long. Whatever the reason, that is a really unfortunate and, I'm sure, a massively disheartening thing to happen. I'm really sorry it has happened, and I'm sorry for playing devil's advocate with this. But I know how instinctive emotional reactions to things sometimes are different to those thought about in the cold light of day, as it were. All I would say is don't be too hasty to assume that she did what she did out of some deliberate intent to make your life awkward, even though that's what's happened.