I'm not sure what you think the difference is between 'gender critical' and 'anti-gender'.
The gender critical movement grew out of feminist discourse surrounding transgendered people, and tends to be fairly centrist or center-left on other issues. The anti-gender movement is older, grew out of Catholic opposition to abortion, gay marriage (and homosexuality more generally) and transgenderism, and is more conservative.
Both are critical of the ideological position that self-declared gender should replace sex (for everyone, not just trans people) in all areas of society, law and policy.
I can't say that I know of anyone who fits that description. I've never met a living soul who thinks laws that protect people from sex-discrimination should be abolished and replaced with laws that protect people from discrimination on the basis of gender identity. If such people exist, I expect they are a tiny minority, even among the most obnoxious activists. In any case, people in both these movements take no care whatsoever to ensure that their interlocutors actually hold this position before voicing this criticism, so it's obviously not about attacking the ideology.
The epistemology underpinning gender identity ideology has its roots in postmodern theory, which is incompatible with science and scientific skepticism.
Can you give an account of the epistomology that underpins gender identity ideology? Or tell me what gender identity ideology is, for that matter? Or how it differs from gender ideology?
Postmodernism is a broad umbrella of critiques of the modernist project with very little in common, so it's more or less impossible to say that it's incompatible with anything. But even those who regard science as a grand narrative beholden to naive realism are capable of doing science. They just need to provisionally accept the grand narrative.
In any case, this is a genetic fallacy, and gender, as a concept distinct from grammatical gender, did not originate in postmodernism but in behavioral psychology/sexology (although certainly it has been contributed to by post-modernists over the years).
Using accurate terms is not expressing hostility to an outgroup, it is expressing hostility to an idea.
"Trans-identified male" is not a more accurate term than "transwoman", and exists only to antagonize those who would like to be thought of as women. "Gender ideology" , on the other hand, is largely a floating signifier used to attack people. You can tell by the way you deploy it against people who don't hold the ideology you (haven't) described.
The approach of TRAs is to frame criticism of ideology as criticism of people in order to suppress dissent and 'win' without needing any argument, debate or evidence. This is characteristic of fundamentalist movements and cults (for example, scientologists framed criticism of scientology as religious bigotry in order to bully critics into silence).
Here we have another example of someone attacking an ideology without bothering to confirm that the person they're talking to holds it. But of course, I'm not merely "framing" the debate this way--these terms have traceable origins. "Trans-identified woman" originated in a sense precisely opposite to the sense that gender-critical feminists have given to it, so it's not remotely plausible that this term was adopted or (continues to be used) for accuracy or clarity. It's also widely regarded as offensive (and not just by TRAs or transgendered people). Continuing to use it in light of this is akin to defending the use of "negro" as more accurate than "black", given that "black" does not exclusively refer to black Africans in English. "Trans-exclusionary radical feminist" was coined as a neutral descriptor, but I don't use it, because it's clearly derogatory.
And yeah, people in cults sometimes disingenuously treat an attack on an idea or practice as an attack on the religion itself or the people who hold it. But it's also true that people sometimes really do intentionally attack others and disingenously foist it off as merely being objective and accurate. And another thing people in cults do is deny that they're in cults. So this is a pretty useless rubric.
You are aware that there are many trans people (especially those with dysphoria who medically transitioned before current trends) who oppose 'queer theory' / gender identity theory, do not explain their experience of dysphoria in these terms, and consider it harmful, right? For example, the
gender dysphoria alliance. Of course, these people get treated by fundamentalists they same way as they treat other critics.
I'm unclear on what you think the relevance of this is. If I think "gender identity theory" (whatever it turns out to be...is it different from "gender identity ideology"?) holds, I will think they are wrong to reject it, and if I don't, I will think they are right to do so.