I guess I could just go with possession. It just might be a little weird writing such a character when the writer is not religious. Basically what she has is extreme black outs, and her fellow students consider her a freak, even though she doesn't remember the event. And her feather is concerned enough to call her parents to take her to her home bunker.
It sounds like you're in for a bit of research either way. Possession would require a bit of research into past possessions, especially if you want to link it to a specific religion (e.g. for a Christian-based possession there'd probably be some references to Satan, whereas a Shinto-based possession might have behaviour linked to a particular Yokai).
Don't worry about writing about religion when you're not religious though. I write about dragons sometimes and I've never met them, likewise I write about Christians and Muslims despite being neither -- I just try to keep my personal feelings out of it and portray the religions fairly.
If you go along the mental illness route there's a whole load of research to be done there on exactly what the illness involves, and which common mistakes are worth avoiding. (I wrote a book about a depressed character with manic mood swings before realising, oops, his behaviour fitted bipolar more accurately.)
Whatever way you decide to go, I'm sure it'll go well -- and you've got the research forum here if you need a hand.
I hope this is helpful!