In my definition, predators are in the coyote/wolf /bobcat/fox/lynx class. Cougars are getting to big game and bears are definitely big game.
So for stuff in that class, I'd use the 223, as the .308 is surely a big game cartridge.
I really like my Remington 223 LVSF. It feels like a bb gun to carry, is nice and accurate to 350 yards (or more, never shot it past there)
To tune it up, if you wish, just bed it, smooth out trigger and you are good to go...