I'd lean toward the .375 H&H. It's not so big that you'll be sent reeling and miss a follow-up shot, and while the .338's certainly not small, neither are coastal Browns.
Also another vote for a magazine rifle. No guarantee you'll need it, but when you're into the alders in the rain or snow with a wounded bear in front of you somewhere, fumbling with loose rounds isn't something you need to have distracting you.
But, something that you can carry. My .375's a very pretty rifle, but it's too long and too heavy to handle well in the brush. Lots of manufacturers now build"Alaskan"-type rifles that look in every way to meet the challenges of the weather and terrain of Alaska.
And, lastly, if you put optics on it, you're going to want the lowest power you can go, and quality to match what you're paying for the hunt. You'll absolutely want the widest field of view possible. The venerable Leupold Vari-X III in 1.5-5x was considered the "standard" back in the day. But, better might be available now.