One thing worth noting is that as a non-resident, you'll be accompanied by a guide on a brown bear hunt.
On my grizzly hunt a couple of years ago, the guides had just come off a month of working on Kodiak, and each just brought those rifles:
338 Win Mag, Sako, 225 gr Barnes TSX
375 H&H, Rem 700, 260 gr Nosler Accubond
I used a 30-06 with 200 gr Nosler Partitions, the other hunter in camp used a 7mm Rem mag with 140 gr Barnes TTSX as I recall. He had loaded 160's but grabbed a box of his 140's by mistake and made a one-shot kill on his grizzly.
However, we were hunting the interior grizzlies, smaller than the coastal brownies. Mine squared a bit over 8' and all the shooting was done at under 50 yards. I remain impressed by big bears. Thought I'd got the "big bear" thing all out of my system... But I dunno… Getting the urge to go again. Might take the 375 if I do.
Guidance I got before the hunt was to bring a rifle with which I was very familiar and shoot heavy, premium bullets. The outfitter was quite content with my choice of the good ol' 30-06 and 200 gr Partitions.
Regards, Guy