Some (dinosaur) therapists will use anything they see as resistance to adopting the gender roles of the gender you are identifying as, as a sign that you are not presenting fully or are somehow resisting your expressed gender role. This one sounds like a dinosaur, though he (I'm guessing the therapist is male?) could also believe that he's being cautious and alert to signs of resistance or ambivalance, given that at least some hormone changes are hard to reverse without surgery or other major efforts.
I'm forgetting which country you are in, Angélique? I'm guessing this kind of chauvinism is still a lot more prevalent in some places than it is in my neighborhood?
You have to ask yourself, practically speaking, whether this is a sign that he has no real intent of signing the letter, or whether you can get past this by submitting, if just for a little bit, to his manipulativeness. In practical terms, if you ditch him for someone new, chances are the new therapist is going to also need more time to get to a point where they feel they've acted responsibly. If doing some chores is the only thing standing between you and the letter, it might be simpler just to take on some chores and "act like a good girl" for awhile, even though it might not be something you want anyone to get used to expecting from you.
There was a time when some therapists would refuse to write letters if a transgirl wore a feminine pantsuit instead of a skirt and ultrafeminine blouse, or if she showed up at a session without lipstick. I heard accounts like these even in the 90s, though they were much more common in the 70s and 80s.