[@FantasyKitten] The simple fact that she has to hold the injuries for some time before passing them, in my opinion, it's enough. Think it like this : She wants to pass over an injury like...a ruptured kidney. It would incapacitate her (or at least, she'll be in great pain), making passing the injury to someone else a great deal of trouble. 

Faint at the sight of blood would be an overkill to be honest. A medic that cannot see blood ? It's something like...a gun without bullets. Useless. 

Maybe...making her have to keep the injury for some time before passing the injury and have the time vary from wound to wound. A simple cut would take let's say, 5 minutes and a punctured organ or things similar would take 20 or 30 minutes. 

Just an idea tho'.