OK, can't help myself, got to say this. Ironglow please don't get offended. Go to the last paragraph and I will give you my current thoughts.
A few years back I was visiting Tennessee. The choir leader at Dad's church came over to visit. Knowing I hunt a lot, and know something about guns, he wanted to get my advise about his gun. Seems he was looking for a bullet that would kill a deer on the spot. Every deer he shot ran off, and he lost half of them. The ones he was able to find the bullets had gone through, leaving a hugh hole on the back side. He wanted to know if a hollow point would do a better job. I asked what caliber he was using.
Now you got to understand, I did not like this fellow to begin with. It was a personal issue, that he had forgotten about, but I never will.
When he said he was using a .243, I looked him in the face and said, "First thing you need to do is get a legal Deer gun". "Get rid of that little varmint shooter". At that time Tennessee had a minimum caliber of .25 for hunting Deer. He had not read the hunting regs very well.
OK, that said, now for my current view. The Nosler 85 or 100 gr partition spitzer bullets would be my recommendation. The heaviset bullets you can get for that caliber. I have taken a couple of girls out using .243s for Caribou. Somewhat larger than Whitetail or Mulies. The Heart Lung shot was great, Bou ran about 50 yards and piled up. The other sister made a neck shot, devastating amount of loss meat. Her bullet hit the spine and totally spoiled all the meat on the off side of the neck. High speed small bullets do that, when they hit something solid, at close range. Better some meat distruction than lose the animal. Throwing the Black Bear in there makes it a bit more interesting. While the .243 will do it, make darn sure you do your part. You won't have the bone smashing ability to drop it on the spot. Advice, if a bear is standing there woffing at you, head on, don't shoot it with a .243. Life could get real interesting real fast.
Yes, I've seen Grizzlies taken with .243s. But the Griz had no idea the little lady was anywhere around. And she made a broadside heart shot with surgical precision. That griz ran over 150 yards before falling, even with a blown up heart.