From my research, a therapist doesn't have to be a "specialist" so long as they are licensed. I had this problem with my old therapist when I was first trying to get on T, and ended up just going to my clinic's licensed psychiatrist to be diagnosed officially. It took me two sessions with her*, and then she gave me my letters for T and top surgery. The clinic itself practiced informed consent, but because I was on the younger side of things, they were wary of giving me T. Ah well, I have it now and I'm actually doing fine. Even though the process took six months or so, it gave me time to think more seriously and look into what I wanted to do for my own health. Sometimes these delays can be beneficial for you in the long run. At least, that's what I like to tell myself when something goes wrong. Panicking makes it hard to rationalize and make your next move.
I know it's what everyone here is suggesting, but if you're older than 18, you might have more luck with an endocrinologist** who practices informed consent. Call around, and be up front with what you're looking for. It can't hurt to try, at this point. You need what you need, after all. And if you can't find someone who practices informed consent who's willing to treat a trans person, then I would suggest looking into a new therapist and being really firm about what you are looking for. You were up front from the start with this one, which is good! Sometimes you just get people who are afraid to write letters should you regret medically transitioning and end up blaming them.
*If you have documented history of trans-related issues, some therapists and psychiatrists will move faster on writing letters and diagnosing you, which can be helpful. The psychiatrist I saw read history from my old therapists, reports from my school for misbehavior relating to the issue, and even spoke to my mother(who is thankfully supportive) to make sure everything was clear and well-defined.
**Endos aren't the only people who can prescribe you testosterone, btw. General practitioners who are knowledgeable can prescribe it to you or refer you to someone who can help you, and to my knowledge, some psychiatrists have the ability to prescribe it for you. Don't feel hopeless if you can't find an endocrinologist, and keep looking.