Well, the problem with your argument for inessentiality, which you pretty much admit in the essay, is that it's observational: this varied group identifies as trans, and you can't easily put your finger on what it is about this group which is different from the other aside from that they say they're trans, and so therefore transness isn't a result of an essential property (not even declaring oneself trans, because one can imagine cases where one would reject the claim).

I think you accept that it's fine to be a realist about gender dysphoria itself. There is such a thing which is sufficiently distinguishable from other things that it warrants its own term, either as a categorical (X-has-gender-dysphoria; Y-does-not) or a continuous (X-has-0.87-gender-dysphoria; Y-has-0.14) variable.

Likewise, one can be a realist about sexual dysmorphia (usually strongly correlated with gender dysphoria), gender euphoria, and so on.

The thing about gender dysphoria is that--unlike gender euphoria which is speculative until the change is actualized--it, if taken in the broader sense that incorporates body dysmorphia, is basically just an observation of being in distress because one is the "wrong gender". It's metaphysically privileged because it is an arrow pointing from one gender towards the other. The correspondence between this and the prefix "trans", meaning to cross categories, is high. It's very hard not to end up entangled in language games in philosophy--and we are entangled here--but aside from that, there's a decent claim for metaphysical privilege.

Is the correspondence so great as to be uncontestable? I think not, but I think the trans-realist position with gender dysphoria as the key distinguishing characteristic has a stronger claim than your essay granted. Especially if you divorce the realist claim from the terminology (which could be its own separate fight), and say: is gender-dysphoria-trans a very clear and real instance of transgender identity? Then I think it is hard not to conclude both yes, and yes-and-it-is-really-a-lot-clearer-and-different-from-some-other-types-of-"trans"-identity.

I'm not sure there's a good nominalist answer to this save to go back and attack the realism of the gender and/or sex categories to begin with, so that the apparently real gender dysphoria arrow is actually pointing in a space where arrows don't really work because there are no universals to point between.

And then you start running into the usual problem with nominalism, which is that it starts getting very hard to keep oneself out of flat-out nihilism...and when you do escape, it's hard not to be actually-at-least-an-anti-realist-but-with-more-sophisticated-mappings-between-reality-and-language-than-the-naive-realist.