I don't see that in the articles you linked. Can you quote the evidence that she equates trans-ness with rape?
I _do_ see something about her questioning whether gender self-identification should affect whether someone is allowed to be tried for rape (since, if I understand correctly from the article you linked, the UK seems to not allow women to be tried for rape). It's difficult to follow to the original quote because some of the linked articles and original sources are paywalled.
Article includes no JKR quotes, but lots of quotes from enraged twitterati. She uses an Orwell quote which doesn’t refer to rape at all, just to doublethink. Please provide evidence, not smear campaigns.
Not sure if it backs up the original "freaking out about..." claim [...]
Not at all. This quote is from a JKR tweet that she made referring to a Times article on the praxis of UK police recording rape crimes as being committed by a women when the perpetrator identifies as female (even if they have male genitalia).
So, neither did JKR (in that quote) equate trans people with rapist nor did she bring up the whole topic. Rather, she reacted to a newspaper article on the topic, and certainly not in a way that could by any stretch of the imagination be characterized as "freaking out".
You are pushing the moral panic in question, and it will age even more poorly than the previous panic since it's targeting and smearing a woman instead of a story.
Did she really state that? If so, please provide a citation.
If not, claiming this would be a grave diffamation that cannot be taken lightly, whether or not one dislikes her position on LGBTIQA+.