FYI, trans woman and woman are the same noun. Transwoman isn’t a word, and the reason people don’t want it to be a word is that making “trans woman” a different noun from “woman” implies they’re not the same thing. Trans women are women, and that’s why the noun is the same. Trans is an adjective.
Yes, you specify that someone is a trans woman by saying “trans woman”. You don’t say “transwoman”, because it’s not a word.