Well, firstly, you shouldn't stretch cold. You should stretch after you exercise or at least done a warmup so that you don't strain your muscles when you stretch. (However, exercise directly before bed will wake you up, so you'll want to find a different time to do this.)
Secondly, a protein shake after exercise = Fine
A protein shake before bed = Fine
A protein shake after stretching = Umm, kinda useless.
Protein is used for muscle recovery and growth, yes. But recovery occurs after muscle fibers are broken down (ie: after exercise) and there is no such breakdown when you just stretch.