What involuntary physiological responses to gender identity can we measure
You don't even need to measure it for pain - you can see it. People signal pain to other people, and we have an extreme response to observing it. We can intensely see and feel pain even in other animals, and them in us.
It's extremely easy to empathise with this. And we can also get our heads around emotional states like anger, love, hate, sadness, and we can have a reasonable stab at identifying them in others, and hence we become able to name them in ourself.
I can also discern the masculinity/femininity difference that tends to align with sexuality, making some lesbians somewhat masculine and some gay men somewhat feminine. If there was a gender identity, it would be that - I can understand the terms "masculine" and "feminine", even if some feminists would deny them - but genderologists are not talking about that either.
All of these things can be seen to the untrained eye. I don't need scientists to tell me they exist. would suffice, and maybe do a better job.
But "gender identity"? I have no idea how to detect it in others - I can see nothing in people claiming "opposite sex" gender identities that I can associate with the opposite sex. I have no reason to think that anyone claiming a gender identity is using the same meaning as anyone else.
So whatever it is it's so subtle that it cannot be clearly detected in others, unlike pain, anger, sadness, happiness. And if I can't detect it in others I have no way to identify it in myself - nothing I can correlate and say "oh, I can see him acting like I do when I feel X - he says that's his male gender identity, ah, so that's what they call that feeling X".
Maybe there is a connection here in that the autistic tend to have difficulty understanding emotions in others, and the autistic are over-represented in the transgender community.
Maybe this "gender identity" seems as real to them as any other emotion - just another abstract thing they can't properly grasp? Whereas to us it clearly does not seem real in the same way as things we actually feel.
If you don't understand emotions or feelings fully, then a made-up emotion/feeling to explain your discomfort with your body is going to seem as plausible as any real emotion.
(Excuse my cod psychology and no real grasp of proper autism.)