Uhhhh, as mentioned already, I wouldn't recommend a 243, 6mm rem or a 257R for bear. LOL, remember a cousin of a gent I shared a deer lease with back in the late 60s. Cousin had driven down to Texas from where he lived in Wisconsin, to hunt our whitetail deer.
He arrived at our lease kinda late at night and brought his gear into our hunting cabin. He sat a box of cartridges on table I was sitting at. It was a big box. Curious, I opened the box and took out a cartridge and stood it upright on the table. Didn't bother to look at the rim to see what it was. I stood a 30-06 and a 308 round next to his. His dwarfed them. I would ask him, just what the heck are you shooting. The cousin, shrugged his shoulders, saying I only hunt bear back home, so all I have is a 375H&H rifle. I kinda laughed, saying yea, that ought to kill one of our whitetail deer.