As long as you hit him right a 22 is more than enough.
A 22 is what all the farm slaughter guys use on pigs and cows.
Not sure if you've seen them slaughter a pig, but after they bleed 'em and gut'em they hang them up and cut them in half right down the spine. You get a nice cross section of the skull, brain, and spinal cord. Anyways what I'm gettin at is that I've seen pigs with skulls an inch thick that were put down with a single shot from a 22.

Don't give the dog away to some shelter. Who knows where it's going to end up next. Yeah whoever they give it too might be good but what about after that. Gotta think about they're kids, and neighbor kids, and friends kids.