How do they know that there is a “mismatch” between their sex and an internal property? What is the internal property. What is the material reality that we can observe that corresponds to this internal property? FMRI and anatomical studies are not able to detect this internal property.
I don’t understand why you say that. The fact that people can be deluded is not an argument against the existence of minds. My position would seem to be insisting that there is plenty beyond thoughts and feelings.
What is gender, in the sense of an internal property? How do we detect it?
I think you fundamentally disagree with this, but I'd say gender is your internal sense of how you align with societal constructs that we associate with certain "gender roles". In our society this is "male" and "female", but other societies have done different things here with the same underlying biology.
Trans people are, and I'm generalizing here, people who feel their biology doesn't match up with the gender role they identify with. They often then want to align their physical presentation with that associated with the gender role, on the belief that societal roles are more important than biology. (It's very transhumanist, in a sense.)
I'll note a fairly easy example of sex and gender differing even in our society, which is intersex people. I.e. those whose physical expression lies somewhere between male or female (which isn't super-common, but certainly happens). They're ambiguous physically, and we historically make them pick (or pick for them at birth) what gender role they'll perform.
How do we detect any self belief or self image? They don't show up on scans. You have to go by reported experience.
If you think that gender dysphoria is rooted in delusion, you should probably read into it. There's a reason trans people aren't treated the same as people with anorexia or schizophrenia - it's because trans people are normative outside of being trans. It's the same kind of situation as being gay.
