I have to agree with kciH about the recoil. However, it does sound like a neat project. I think there's a riflesmith by the name of David Clay that converts the 1886's to 50-110. I have no idea what his prices are, but the ones I've seen pictures of all wore muzzlebrakes. I bet there's a reason for that, too. I have been waiting for Winchester to bring the 1886 back out in the 50-110 chambering. I'd sure like to have one.
Don't let Alaska cause you to do something you don't want to do. A stout 45/70 with proper loads, either the newer 1886's or the Marlin 1895, is more than adequate for the big bears. I would feel perfectly at ease with the M71's original chambering in 348 Win. Another rifle to consider is Winchester's M1895 in 405 Win or send to Z-Hat for one of his Hawk conversions. I've been working with 411 Hawk in a Winchester 1895. Here's a thread on it.
http://shootersforum.com/showthread.htm?t=5868