I would think that if you have anything at all more powerful, then I'd use it. I know that plenty of folks have shot dogs/people/cats/deer with .22lr so they claim. Personally I am NOT AT ALL interested in wounding an animal so that it escapes initially, only to bleed out later where I can't find it.
I'd spring for a .223 or a shotgun, or even a .17 Remington or a .17HMR before I bothered actively hunting bobcats or other predators with a .22lr.
Just my $.02