Withholding food for 12 hours won't kill them, but water should be given to within a very few hours of surgery. The reason to withhold water from animals (including human ones) for a few hours is to prevent aspiration of vomitus into the lungs. In adult humans we often say 12 hours, because we adults don't dehydrate quickly, and hospitals try to error on the "safe" side, since most patients "cheat." For babies and toddlers, 5 hours of no fluids is good. Your chickens won't vomit and aspirate. (Neither will rats.) Dehydration is VERY bad for surgical patients of ALL species, and you will have much better results with a well hydrated bird. Stress on top of dehydration is just asking for trouble.
If you are trying to dehydrate a bird enough to decrease the chance of bleeding, you are REALLY compromising his chances of survival. Vascular collapse, heart rhythm irregularities, etc.. Listen to the vet.