Lose some cattle to stray dogs and lose a cat to feline leukemia from a stray cat and it makes you kind of pragmatic. Also, the .308 is awesome for making shots at 200 to 300 yards. I know it is overkill, but with the .308 you can make one shot and know the critter is dead instead of suffering like they can with a .22.
Believe me, there will be no suffering on a cat sized critter with a .22-250. It's a great centerfire varmint round. If you hit it, it's RDRT. Very flat trajectory, too.