"If given the chance, players will optimize the fun out of your game" applies here XD
Totally agree that the movement makes it feel alive, and I wouldn't wanna sacrifice that either. Maybe since the interaction is usually so small (i just want THIS bird and THAT bird to get SLIGHTLY closer, usually in the corner) you could do something like giving the player a "blank" round that they can fire with right click, that has a small radial push effect, which they then can follow up with a real shotgun blast. That'd give us a little more agency/another interesting choice to make/another upgrade to offer.