Indeed. Also, a .556 round doesn’t exist. AFAIK. The NATO 5.56 is similar to the Remington .223, which is slightly larger in diameter that the ubiquitous. 22LR plinking round, though delivered at MUCH higher velocities.Im no expert by any means but me thinks you might want to up the caliber a smidge.
It could definitely kill a bear, particularly a black bear. Eventually. But it’s not a “stopping” round. You want a much larger, hard fast flat nosed round for that.
The debate about what’s appropriate rages on from there.