I wouldn't go hunt cape buffalo with a 300 magnum just like I wouldn't hunt brown bears with a .308. Just the same reason that a .375 H&H is the legal minimum in most of Africa the .338 should be a personal minimum for the large bears. Basically if you are going to hunt dangerous game then you need to step up to the plate and learn how to shoot a heavy recoiling gun or stick to non dangerous game.

The .308 will kill anything that walks this earth but there are reasons that larger cartridges were made.

If I were (I wouldn't) to hunt brown bears with a .308 I would use either trophy bonded bear claws, Barnes X bullet or Swift A-frames all in 180 grains.

Honestly if you have the money for a new rifle and if you can handle the recoil and shoot a heavy recoiling rifle accurately then I would step up to a .338 Winchester mag, .338 Remington ultra mag, .350 Remington mag, .375 H&H and other similar cartridges.