Hell, just about anything will kill Black Bear, even very big ones! The reason you traded the 7mm for a 308 had nothing to do with killing power, but was because of the left handed shooter! The key to killing a black bear without loseing him is place the bullet in the right place with the first shot. The 7mm Rem Mag has just as much killing power that the 308, from muzzle to 300 yds, or farther. The key here is to use the right bullet, and place it where it does the most good. The 308 is a fine black bear cartridge, but so is the 7mm, but your advantage is haveing a rifle you can shoot properly, in the BLR.
A lung shot on a black bear, is sure death to that bear, but many are lost with lung shots, because the black bear tend to not leave a blood trail. This is because the skin is loose, and there is a layer of fat under the skin. This causes the bullet holes in the skin on the enterence side, the fat layer of that same side, and fat layer on the off side, and the skin on the off side to not line up while the bear is running, so no blood trail. A Bear cna sustain horrendous wounds, and leave no blood trail at all, but bleed out in a short distance internally, and die! Top this off with the fact that most good bear habitat is in moss covered floor woods that soaks up any blood that is dropped, makeing it very hard to track them.
As was said earlier, one must turn off the electricity to drop him in his tracks, and the only thing that will do that is a CNS hit, ot a shot very near to the CNS, but even then when he hits the ground tap him again, reload, and watch him very closely, because they sometimes get up after a while, so be ready to shoot once more!
A 243 with a proper bullet, and placement, is just as deadly as a 500NE,on black bear, or anything in between, if the right bullet is placed properly!
.................Good hunting with your BLR, and I suggest a 180 gr Nosler Partition, in front of 43.5 grs of H380 @ 2360 fps! Place it right, and you got a dead black bear!