Long story short, the best thing to do is some zen deep breathing and just... Ignore her. Treat her civilly like everyone else, be friendly and include her, but when she says weird shit, just blank her. Act like it didn't happen. She does it for a reaction, and when the reaction doesn't come, she'll (so, so slowly) learn her behavior is pointless and makes her look like a fool.

Why do all this? Why not just disqualify her from potential membership in the sorority in the first place? If she's already in like Flynn just coz she expressed an interest in joining, what's the point in having a selection process and pledge period meant to give the group time to get to know and take the measure of the recruits?

I wasn't in a sorority, but I was in a "senior society" in college. AFAIK sororities, like other private clubs, don't have to let anyone and everyone in - they are supposed to be exclusive clubs that only take in women who meet certain criteria. And the minimum criteria used to be that one had to be sociable and both acquainted with and capable of following the rudimentary rules of decorum - which would eliminate anyone as anti-social, unsocialized, inappropriate and IMO mentally unwell as this woman.

Or are sororities now just pathology-enabling societies?

Besides, she probably only mutters "kill TERFs" because her only friends are on the internet, and that's what she has to say to keep in that crowd. Once she starts making real life friends and developing positive interests (Idk what sororities do, but probably not sit around tweeting about depression yes?)...

Sorry, it sounds like you see joining a sorority as akin to going to a mental health facility. From what I know of sororities, somebody who is in college and yet has never made RL friends or developed any positive interests would not be considered a catch for most. She wouldn't even meet the most basic criteria.

Besides, she probably only mutters "kill TERFs" because her only friends are on the internet, and that's what she has to say to keep in that crowd.

You're making it sound like she is a feral person raised by wolves on the internet and therefore she can't help herself, and so anything goes from her, and her vocal advocating for the beating and murder of women must be tolerated and tip-toed around.

But clearly, if she's a college student at a regular college or uni, she must have grown up in some kind of household, perhaps a family, run by elders, and she must have gone to school for many years and done well enough to graduate - so by now she should know the basics of how to behave in civilized society, be aware that assault and murder are morally unacceptable as well as criminal, and be capable of handling "nuanced conversations."

If this woman has special needs, so be it - but then she needs special help from people trained to provide it. It's not up to OP or other members of her sorority to do this; they are not mental health care providers, special ed teachers, or professional carers.