Lots of caveats built into your question. For our purposes, can our pistol and shooter deliver the shot? Then will a .177 pellet penatrate a pigeon' s skull up to 15yds? Sub <8gr pellets 2-3ft/lbs energy, which likely won't knock 'em out or penatrate. I shot rats, some big, with a CO2 .177 pistol a few evenings and think they'd be similiar. Perfectly placed, direct not glancing angled, head shots were consistent kills inside 5yds. (I had a cheap trigger and couldn't place shots any further.) Some ran off.

Can you kill a pigeon out to 15yds? Can you hit a spot the size of your fingernail time and time again within 1 or 2, at the most 3 sec.s to setup and aim? (unless your slipping into the barn and getting 'em at night) Maybe can kill one, but guessing you'd see more fly away with a bad headache or die painfully elsewhere.

These questions get asked alot, so people have a need to know. If you try this shot, you'd do a good service by getting back with a post as to what you learned. Folks reading this forum could then get an idea of what low power guns can deliver in the way of humane kills on small critters.