I can totally relate to the urge to shoot that hawk. Yeah, stifle the urge to kill it, but you can get close as you can and let loose with a loud round over its head. Sudden loud noise can often deter a hawk. Repeat the close-by gun noise, and perhaps the hawk will seek a more peaceful environment.
BBs may not kill, but they can cause sickness and eventually organ failure as the BBs circulate through the hawk's body. I know this because I had a good friend whose cat became sick, they took her to the vet, and the vet found BBs in the poor cat's organs. A crabby neighbor from their previous neighborhood had been shooting their cat for years before they moved away. BBs are not harmless. They often don't just bounce off the animal.