7.62 x51/x39
The 7.62x51mm Nato (commercial name .308 Winchester) will throw a heavier bullet nearly 20% faster than the 7.62x39. This translates into a substantial increase in energy. I don't know if it would be enough to satisfy your desires, only you can decide that.
However, the .308 is more than adequate, in skilled hands, for North American hunting. Actual defense against bear attacks is different than hunting, and most people prefer bigger rounds for defense, but a .308 had all the energy needed, if you apply it in the correct place.
Be aware that while the 7.62 NATO and the .308 Winchester are dimensionally the same, commercial ammo can be found loaded to slightly higher velocities and pressures, which may not be a good thing to shoot in military rifles with oversize chambers.
The largest grizzly bear on record (1901) taken in the lower 48 states was taken with a .30-40 Krag, which is slightly faster than the 7.62x39, and uses much heavier bullets, but falls below the level of the .308 Win.