So I will use the language “female born woman” when it is necessary for some reason to differentiate them from transwomen.

Or, just a thought here, you could use the commonly-accepted term “cisgender women” or “cis women” for short, instead of insisting on bulky phrases that, even with a nominal concession to sex/gender difference, still manage to convey the sneering sense of “not real women” that the previous “women born women” terminology has traditionally carried. “Cisgender” is a term mirroring “transgender”, with exactly opposite meaning; where “transgender” means an individual whose assigned sex at birth does not match their gender identity, “cisgender” simply means an individual whose sex assigned at birth and gender identity match. A quick snippet from what I’ve found to be the best writing on the subject of marked case, unmarked case, and why we should use trans/cis terminology:

(‘cisgendered’, from ‘cis-’, “on the same side”, + ‘gender’; in contrast to ‘trans-’, “crossing over”; both from Latin, and both prefixes used in Chemistry with similar meanings.)

The reason it feels jarring — “naming”? “marking”? — to you, and gets your hackles up is quite simply that y’all are accustomed to being the unmarked class, and giving you any concise name is going to feel like an imposed label that, because you’re not used to having to acknowledge a label at all, some of you start to suspect is somehow insulting or denigrating.

Also, if you are trying to be taken seriously as *not* anti-trans (and I am doing my best to assume good faith on your part, despite the fact that the “women born women” segregation you are part of and defend absolutely screams “transphobic bigotry” to me), I *very strongly* recommend you write it out “trans women” not “transwomen”. Here is an excellent post on the subject, and how writing it “transwomen” is inherently Othering because the counterpart to “transwoman” is almost never “ciswoman”, and thus it positions trans women as not women, but “transwomen”, something other than real women. (In fact, I used to write it that way myself, until a trans friend of mine called me on it and linked me to that article; I hope you will find it as illuminating as I did.)