I think @DadJoke might find it helpful if he went back to treating "trans" as a prefix (which was how the TRAs originally treated it - remember that battle cry "It's a PRE-fix you BIGOTS! It's yer actual LATIN!* Have you never heard of TRANSALPINE GAUL? That's why you other lot of people who aren't trans are CIS, because the other way was CISALPINE GAUL etc etc etc")
As a prefix (even though we didn't accept "cis" for ourselves) it modified the word woman. Prefixes are most commonly employed to make words into their opposites eg un + happy = unhappy; im + mature = immature; il + logical = illogical. You get the drift.
Then, almost by osmosis, and in the dead of a dark and foggy night, "trans" began to morph into what at first glance appears to be an adjective - TRAs did this by separating the prefix from the noun. And then they got annoyed when the "transwoman" was used instead of "trans woman" - even though that is what they had always done themselves until that dark and moonless night.
Suddenly "trans" seemed to be an innocent adjective, so a "trans woman" was apparently (in the addled minds of the TRAs) just as much a woman as a tall woman or a thin woman or a blonde woman. (SPOILER ALERT: no way is it an adjective; no way is a transwoman a woman). But it wasn't innocent at all - nor is it an adjective. It is an archaic word, normally used only in science (Oh! The irony!) which has been slipped into everyday language under the guise of "being kind" and trying to legitimise a male fantasy.
This may be how @DadJoke got confused. If he goes back to using "trans" in the correct sense (eg as in Transalpine Gaul - the Gaul on the other side of the Alps) he may find things easier. Transwomen are on the "other side" of woman - ie men.
*Latin is a very scientific language. Also a legal one - 2 stones, 1 bird etc
** Yes - I know I wrote 2 stones, 1 bird. It was intentional.