They believe that if anyone can ‘become’ a woman, then the word ‘woman’ means nothing, and therefore, apparently, trans people are out to destroy the entire concept of womanhood by their very existence.

That strikes me as the opposite of the TERF position, which I understand to be that (i) anatomy is the only difference between the sexes, and (ii) anatomy alone should not be socially, economically or poitically relevant. One can disagree with the second point, but that issue is moot if there are other differences, and transsexuality implies that such other differences exist; if they didn't. transsexuals wouldn't "know" that they are trans. So, yes, trans-exclusionary radical feminists are transphobic because they fear, rightly, that transsexuality debunks their most deeply held article of faith - the absence of non-anatomical difference.

I would argue that transsexuals are the real essentialists, but for them, the essence of their sex is not anatomical. The anatomy usually comes with the territory, but sometimes it doesn't. Either way, one's sex is the sex they know is theirs, precisely because that sex has an essence they can sense, regardless of their plumbing. Transsexuality is thus an attack on the lack of subtlety in the radical feminist worldview.