I posed the question to The Man (works with guns all day every day), and he mentioned an air rifle called "Whisper." He said he wouldn't rely on it to be a clean kill (or kill at all without good placement), but it would be good enough to detour and control if that's your only option.
Good for you. You're a good shot. The OP might not be, hence the need for something more powerful than an air rifle.
Personally, I would never attempt a one-shot kill on a larger pest with anything less than my .45 lever action. I've seen possums take three 9mm rounds at point blank and keep...
As another poster said, check into your county ordinances. Most have a clause about killing nuisance animals, anything that "harasses" livestock. If your county has that, use an actual rifle that's reliable for a one-shot kill. If your neighbors pitch a fit, cite that ordinance. I'd even keep it...