I have used the Federal Blue box 150gr. .308 for 8 or so whitetail and mule deer. Based on penetration and expansion that I observed, and the two bullets that I actually recovered, I would feel confident recommending them as OK for black bear.
But if it was me, I'd use 180's to be more certain of an exit hole, which is very handy when you need to track a bear that didn't die quick enough.
I would not put the Sierra Game King in the same category with Accubonds or partitions. Game kings are accurate, but no better at retaining weight or penetrating deeply than the Blue box federals. If I was buying ammo for my .308 to hunt bears with, it would certainly be 180 grain, and probably Nosler partitions or Accubonds. The $10 more for a box of shells would make me feel better, but probably won't kill most black bears any better than your plain 150's.