Magnum Wheel Man
Having lived in Kodiak and encountered a few Kodiak Coastal Brownies, I'm too chicken to risk it with a .270 Win too...
My standard bear gun is a Marlin 1895STP in .45/70 loaded with 405gr hardcast lead pills. However, I've carried it mostly for defense purposes though I also carried it for "backup" purposes for my buddy who took a Kodiak Brownie with a bow without incident.
IMO, the last thing in the world that you want when hunting a big brown bear is a whizzum-bang-earschplittenloudenboomermagnum. I have a personal distaste/disdain for the .300 Win Mag, it won't do anything a .30/06 or .308 won't do in a realistic hunting situation with proper ammo. Just because the same weight bullet is going faster doesn't mean it'll kill a bear quicker, and the increased speed is actually counter-productive with all but the toughest bullet designs.
The .338 Win Mag is one cartridge that I'm on the fence about. It's fairly well-rounded for just about all AK big game, assuming proper ammo selection and ranges. But it still just kind of "irks" me as a brownie gun because of the high velocity and negative impact that can have on less than ideal ammo selection.
The .375 H&H is about as close to the perfect AK big game cartridge as you can get. This is especially true if the shooter is a handloader. You can get .30/06 performance for deer, and you can get full-house .375 H&H elephant-gun performance for the more dangerous critters. If I could only have one centerfire rifle for AK, it would either be a .375 H&H or a .45/70.
The .416 and up class of "African Safari" cartridges will work fine on a big brownie, but are a bit overbore for the average hunter and are way more gun than what is necessary for smaller, less dangerous species of big game such as deer, elk, moose, etc. But hey, if you have one, and you can shoot it well, why not?
The .45/70 is an ideal brownie hunting cartridge. Quick handling, relatively light, and shorter rifle length compared to many others I've mentioned. I prefer it for brownie hunting and for defense. Most shots on a brown bear in AK will be less than 150yds, and the vast majority will be less than 75yds. Either of these ranges are well within the capabilities of a .45/70. Most guides I personally know in Kodiak carry one variant or another of a .45/70 in a Marlin lever gun. That speaks volumes to me.
One poster mentioned that a Kodiak Coastal Brownie hunt was on their "bucket list". Which brings me to a major point... For most people, an Alaskan bear hunt is a once-in-a-lifetime opportunity. One that costs as much as a new crew cab pickup, and one that you will remember until your dying breath. Don't jeapordize that by choosing an insufficient cartridge, nor by choosing a cartridge/gun that you are not inherently familiar with/can't handle well due to any factor, recoil included.
For non-residents, you will be required to have a guide. You should contact them at least 8-12 months prior to your hunt and seek their advice on everything concerning the hunt. This conversation should focus as much on guns/ammo/shooting techniques as it does physical conditioning and excercises that will help prepare you for a remote Alaskan hunt. Both are equally important. You don't want to have a heart attack on the side of an Alaskan mountain (though I can think of worse places to kick the bucket).
I didn't include a lot of people's favortie cartridges in the above list for one of two simple reasons. Either I haven't seen the cartridge (such as .375 Ruger) widely available on ammo shelves in Kodiak, or it's too far on the light side for the intended hunt, IMO (a perfect shooter with an accurate, familiar .30/06 loaded with 200gr Partitions would be the lowest I would go).
Any questions, post them here or PM me.