the claim that trans people have some "internal representation" of gender that is inconsistent with their biology

This is (obviously) a disputed proposition. It's certainly the minority position within modern psychology. Particularly since a fairly sizable number of trans people will report actual body dysmorphia - a belief that their body has the wrong genitalia and secondary sex characteristics (like hair and breasts) - rather than merely distress caused by the social and cultural aspects of gender. Since virtually every society has gender norms, though, there's no "slam dunk" way of demonstrating whether the physical dysmorphia is interrelated with, or independent of, the cultural aspects. We know that purely physical dysmorphia exists ("My arm shouldn't be here"), so it's certainly within the realm of possibility that trans identity is no socially constructed.

Fortunately, we don't need to wade into that incredibly sensitive question. The reason there's no "race dysmorphia" as a pyschological condition is that it simply isn't observed among patients. You don't see a non-trivial number of individuals reporting mental distress and trauma resulting from an inconsistency between their internal sense of their race and their external presentation of their race. Because one is an observed psychological state that is reported by a non-trivial number of patients, and the other is not, one is considered a psychological condition.