I don't think there's a simple answer to this. Offhand I'd say it hinges on what we take to be 1) the objective reality that exists irrespective of us (being aware of it), and 2) what we take to be the cultural, ideological etc. forms of how it get represented.

For example, take some law of physics. People who subscribe to modernism probably agree that it describes, or at least very closely approximates the objective reality in some crucial and relevant way. But it's still a cultural form of representing that reality. Without denying that reality or implying there's "other ways of knowing" or any other relativistic stuff, we can still think that, even in physics, there could well be other ways to approach the same thing, and maybe we don't know them all. Here's Max Tegemark, a professor of physics in MIT talking about it with theoretical physicist Sean Carroll:

"...you also take seriously the idea that the laws of physics might have more than one solution for what a uniform space can look like. We know that’s the case for water; it can be solid ice, or a liquid water, or steam. String theorists say the space we live in might be like that too, except there might be more than three ways… Maybe more than a googol ways it could be… This is a pretty general phenomena in physics. If you have some complicated equations, they can have multiple solutions." (Sean Carroll's Mindscape podcast).

(Just for the record, it's probably safe to say that most of what they're talking about goes way over my head, so I'm just trusting their authority.)

Of course, epistemology (what can we know, and how do we know it) and ontology (what is real, what exists) are ancient philosophical topics on which there is no agreement. One just has to take a look at the options and provisionally pick one that seems the most reasonable, and then adjust one's beliefs if there's need.

When people who I think are "extreme social constructionists" talk about stuff like homosexuality or heterosexuality, they might say that these things were invented in the 19th century. In some sense they would be right, but only to the extent that they're talking about the cultural representation of the objective biological reality of sexual reproduction and the "drives" that make animals (homo sapiens included) do it. Or, to repeat an example I've made earlier in another thread, like Alan Sokal and Jean Bricmont wrote in their book Fashionable Nonsense about Bruno Latour's musings that pharaoh Ramses II couldn't have died of tuberculosis because the germ that causes tuberculosis was only discovered in the 19th century.

Things don't divide easily into obvious categories of objective reality, a cultural representation of objective reality (heterosexuality, sexual reproducion), or a cultural phenomenon that is part of social but not 'objective' reality in the sense that the latter would follow from the former (what it means to be a mother). That's why I think it's a legitimate discussion (if you have it in good faith etc. of course) whether or not "gender identity" in some loose sense (the awareness and experienced importance of one's sex and the cultural meanings attached to it) might be "transhistorical", in the same sense that heterosexuality is, even if these are relatively new concepts. (This is not to say anything about the conclusion of that discussion.) When you say,

I don't think we should label historical figures trans or non-binary when there is no way they could have thought of themselves in those terms.

I tend to agree, but I'd still say it depends. Some time ago I listened to an interview about a historian who had looked into the practice of spiritism in the early 20th century. Her approach as a historian was that she didn't want to look at the phenomenon from her personal vantage point in the present ("well of course it was obvious nonsense, and scam artists made good money off gullible people"), but from the point of the people who were involved in it, who were commenting on it etc. (some claimed it's real, others believed them, others weren't so sure).

I think it's a legitimate point of view, even though if I were a historian, I wouldn't use it. With all this relativistic "different ways of knowing" stuff being so popular as it is, I'd "virtue signal" by keeping a visible distance from it. But I still think it's a legitimate point of view, given that you're clear about what you're doing, and don't slip into de facto peddling superstitition etc.

When you write,

We can of course talk about GNC and third gender people but this should be done in a way that is cognizant and respectful to the cultural context they existed in which does not include twentieth-century psychological theories of gender identity.

I agree. Probably it's impossible to have conversations on Twitter about this, and indeed there "[it] is simply a propaganda device to establish a historical basis for trans ideology that doesn't actually exist", as you say.

And TRAs aren't the only ones to do it: nationalists of all kinds want to think of themselves are the heirs of some ancient civilization, socialists partly legitimate their political aims with primitive communism ("see, it's kind of been done before"), see antecedents to it in certain sections of Plato's Republic etc., and while I don't recall right now having come across any feminist texts doing this, I'm sure they've done it too.