.308 is way too much gun for a cat. A .22-250 is more than enough and shoots flatter in the bargain.
Besides, it'll turn the cat inside out which proves that there is more than one way to skin a cat.
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.