The question is not so much what you’ll be hunting as, will you be in bear country? I have hunted caribou in Alaska with a .270, .270 WSM, and 7mm Weatherby Magnum, and all three did fine. Except that, on the hunt where I had the 7mm, I was checked out by a young boar grizzly, who seemed to find the guide, my friend, and me mildly disappointing and wandered away. If he had been a mature boar grizzly, I might have wished for a much bigger rifle.
I’ve known, personally, two guides who had to kill bears (one a brown, the other a grizzly) who were trying to do the same to them. One guide did the job himself with a .416 wildcat. The other guide had a .44 Magnum revolver, and the attack took place very suddenly over the disputed carcass of a caribou. The guide told me that if his client had not stood his ground and shot very quickly and very accurately with a .338, he might not be there to tell me the story.