jweiss,please do not punish your first cat by giving her to HS,she was there first
I too have a cat who is not too fond of others,but now he tolerates my other cat,he just cannot get too close.
You have gotten some good advice already,but if Bianca is getting sick and vomits,although she has a clean bill of health,it's very probable that she's stressed,stress can cause all kinds of problems.
Having no claws at all,back and front,she might feel helpless and has trouble getting up high away from the others.
You owe it to this little girl to make sure she gets some relief,preferably in a room on her own.
PS,I am happy you will not declaw any more cats,it is a horrible procedure.