It's useless to try to educate bigots. Citing the statements of authoritative bodies is lost on these know-nothings. They lack all ability to understand nuance. To them, it is impossible that a completely sane person born with a penis can think and feel and self-identify as a woman.
But let's try:
Imagine a completely normal woman who awoke one morning to find that her female body had been replaced with a male body. Yet in every other way this person continued to experience the world as a woman. If this could actually happen, would you call this person "insane" for continuing to insist "I'm a woman"?
Or is it possible that a totally sane person can look at their body and their feelings and say, "I can see that I have a penis, a square jaw, a muscular physique, and lots of facial hair that needs to be shaved every day. I can see that in every physically-detectable way I have a man's body. But my body doesn't define my gender. In every psychological way that a person knows whether they're a man or a woman, I know that I'm a woman."
Well, this is exactly the state that transgender people find themselves in: their perception of their own gender is exactly the opposite of their biological gender. And professional bodies who assess mental illness do not list being a transgender person as a mental illness.
But of course, the right-wing know-nothings have an opinion. And because they just aren't capable of "getting" what a transgender really is, they make the gross oversimplification that it's just "cross-dressing" and insist it's a mental illness.
If all of the right-wing bigots were replaced with transgender, the world would be a lot better place.