My recommendation is more general and not confined to a specific cartridge.
I would look at big bores (416 to .500) because of there knock down power. Brown bear are
very dangerous and you need to put them down hard with one shot. big bores will do this very well.
There are lots of choices and I would make sure I had enough rifle. if you have time, look for something that will sell after you make the hunt so if you never go again you have not bought something that you will be stuck with. stay with something like the 416 rem, 416 Rigby, 458 win
very capable but not a safe queen. Also ammo can be bought more easily.
The 375 H&H would be the smallest I would use and even though lots have been killed with smaller
cartridges rarely do they go straight to the ground.
For a once in a lifetime hunt, you should go with more than just enough rifle. There is no such thing
as over kill when it comes to something like brown bear.
Just my opinion
J E CUSTOM