Except for the biggest beasts, I think the 338 cal could be about ideal. Variety of bullets, a few chambering choices(I'm hoping for one like 338 Lapua in my next barrel) and they are usually accurate.

Stepping up to 375 is not a huge step. If I'd be inclined to need something bigger as a stopping round, it would have to be 40+ cal.

My buddy is about to start guiding bear hunters next spring in Alaska. There I think you need nothing less than 458. But if you are simply hunting them then I'd feel ok with 338.

PS on a few forums they use the sierra match king bthp bullets in the 338 lapua. 300 grains, accurate and they claim they work like a hunting bullet, not like a varmint bullet. At least its another to consider.
I used 225 X last fall in 338 Win to nail a caribou at 802 and the bullet penetrated and expanded. What more can you ask? 2nd round took out the spine and never slowed down.

Jeff