Delusional, no. Unwise, yes. Legal things should *always* be in your legal name. I think the term is fraud but I'm not certain. Still, depends on the legal documents you signed to get the card in the first place I'd imagine.
My advice: Make sure you do not give
anyone any excuses.
As far as delusional goes, I've not heard anything that I would call delusional.
Being transsexual can *easily* be called delusional. Look at it from the perspective of thinking being a man or being a woman is black and white and there are
no in-betweens. If I were to tell such a person that I was a girl how possibly can they compute that. That I am delusional is the only logical answer.
Moral: Understand why someone tells you something before you believe it

Judging by what you have said of the therapist, it would not by chance be a traditional sort.. older possibly. Believe marriage is man & woman at a church only.. that sort of thing?