Transitioning obviously being the easier option since us trans girls are so universally loved and respected.
I can get it on some level, growing up surrounded by homophobia (but not transphobia as your parents never even heard of the concept), so being gay is bad and I'm not bad so I can't be gay. Then discovering the concept of transness, seeing it as an "out", proving that you weren't bad, there was a third option, so this must be real.
Psychologically this is valid, and I can imagine a psychologist coming from a good place trying to figure you out entirely before assigning a label that could result in more issues (like you said, it ain't easy being trans).
Of course he could be a trans denying asshole too, that's hard to tell from text.
And there's obviously better phrasing to explore the possibility of you being gay instead of trans
In the second meeting we had I pointed to my genitals and said that these have to go. Dude looked super annoyed but that was it, the meeting was over and I got my referral.
That sounds like the exact thing you should do, that's not something that can be denied (maliciously or otherwise) through internalized homophobia.