If you can put a .22LR into a critter's ear standing broadside it should suffice. An Inuit woman killed the 1953? world record grizzly that way. It was trailing her, and she moved around some big rocks for the broadside shot; it worked!
Wowza...getting followed by a grizzly and carrying a 22...I would need a fresh pair of pants after that one!
But interesting you can hunt deer in Texas with 223/556. What's the smalls caliber allowed there?