How is that over thinking? A small shot of morphine will dull pain and make you feel fuzzy. A somewhat larger shot will knock you out and a big one will shut your lungs down and kill
you. Whatever is in those guns, at a certain point, there's bound to be a lethal reaction.
Did she grab the icer because she didn't intend to kill, or because it was the only weapon to hand and a safely unconscious person is much easier to kill.
Look at it logically. What was her end-game intent here? Fitz found something he shouldn't have, so she knocks him out...and then what? If he comes to then she's still outed. The flip side of that of course is what to do if she did manage to kill him? Stage an accident and hope the others buy it? Dump the body out a hatch and play dumb?
One thing is certain though, there's more going on than we know here. Personally my money is still on her being Fury's eyes and ears.