[will a 6mm be ok? We're talking 200 yards and closer.
If I can use a 6mm, please explain the best bullet to use as well as what grain of bullet to use.
AlphaDog, to a guy who has hunted black bears with a 6mm you pose a hard question to answer, especially since you've said you will be hunting with your 14 year old son. I know three or four guys I'd be comfortable with hunting black bears with a 6mm. They are hunters of long experience on many types of game including lots of bears, cool disciplined good shots whom I know will pass questionable shots and only take the sure thing with the 6mm. If one of you has the .270 and one the 6mm, you take the lighter rifle and give your son the .270 if he is comfortable shooting it. Otherwise, stay close and only let the young guy take a pretty close shot with the 6mm (preferably 50 yards or less) at a whole bear standing still, unless your 14 year old is a lot more mature than most of us are at that age.
Basically, I wouldn't recommend that you hunt bears with the 6mm. But that is what you have in your arsenal it sounds like so...
I'd suggest you keep all shots with the 6mm to under 100 yards, for you as well, and shoot from a rest so you can precisely place the bullet. Broadside through both lungs and he is yours, though it may take a few seconds and bears do not leave blood trails sometimes from smaller wounds. A 95 or 100 grain Nosler Partition should break or at least penetrate the shoulder into the vitals. The most thoroughly and instantly dead black bear I ever saw was a large bear shot at about 25 yards with a 6mm pushing a 95 grain Nosler partition. The hunter sat on the edge of a steep drop-off and shot the bear between the shoulder blades as it fed on a swampy flat below us, head down, facing away. The bullet entered the back of the withers and followed the inside of the spinal column the length of the neck into the brain and stopped in the skull. The bear dropped with a thud on the spot. Don't count on that kind of shot opportunity or lucky placement.
I've never seen a black bear charge a hunter after being hit or shot at but it could happen. (I have seen them run straight at a hunter, like one guy who thought he was being charged, but the bear was running from the echo of the shot on the far wall of trees, not aware the hunter was there and veered away once he saw the man.) I'd be much more concerned about following a wounded one than about being charged or attacked immediately upon the shot. Therefore the critical thing is not neccessarily to anchor him, as I'd prefer to do with a grizzly, but to insure that the first is fatal within half a minute or preferably much less.
A 100 grain Nosler partition is a good bullet in the 6mm.