I agree with this. Also think that ED speculation is highly unlikely as there's no reason for an elaborate cover story. Considering all the work they do with MH charities and with Harry & William being open about their own struggles, it would be downright bizarre to hide an ED like a shameful secret (which I don't think she has anyway).
I think the reason for the hospital could be that she's with a specific surgeon/consultant who is using their facilities for the surgery. Private hospitals are far more flexible with whom they work with, and for a patient of that calibre, any doctor can come in to use their equipment.