Major Khan,
first, thanks for all your great stories and great information of a time gone by.
I have lived in alaska most of my adult life, hunted here and guided for bears (brown/grizzly) and have formed some opinions, i will try to convey them.
I have carried and used 338 win, 375 ruger, 45/70 marlin and a 450/400 double rifle on grizzly/brown bears. I have shot bears with all but the 450-400. I have seen clients shoot bears with 300 ultra mags, 338 ultra, 338 lapua, 458 winchester mag, 375 ruger, 375 h & h, 375 weatherby. the ONLY rifles that i thought did a poor job were the 300 ultra's. too fast, no penetration. multiple shots (more than 5 in some cases) were always required. they were using swift A frame bullets in every case.
the double rifle is largely ignored as stated many times before, cost, availability AND BLUED STEEL. when i guide brown bears, my rifle WILL BE COATED WITH SALTWATER EVERY DAY! getting off the boat, on a skiff, hunting, then returning, it will be coated with saltwater spray and likely be rained on almost every day. i cried a little every day when a hunter with a 375 h&h sako bolt rifle (blued/wood) got back on the boat in the evening. soon the stock turned a nasty pink color due to the finish on the stock absorbing water. i cleaned that rifle for the hunter 3 times during the hunt (10 days) and it was rusting up each time.
there are two kinds of rifles for the grizzly/brown bear hunter. the hunter/client rifle and the backup/guides rifle. i think a 375 and up of any type is a fine backup rifle, even a double would be fine if you were willing to subject it to the weather. the guide needs a stopping rifle, so i believe a 9.3x64 and up would be appropriate. the client needs a rifle they can shoot well so a 35 whelen power level rifle, 9.3x62 and up would be appropriate. i would rather see a hunter show up with a 3006 and 220 gr bullets than a 300 ultra mag!
i have a 450/400 zoli o/u double rifle that i bought for all around use, have shot several moose with it, no brown bears yet.

my hunters shot very well when i carried it and was not needing a backup shot, oh well. i wax the rifle every day when i get off the skiff and back on the boat, oil it as needed, etc. it is horrible conditions to bring a beautiful rifle to, that said, I'm gonna bring it on this spring hunt.
i spoke to a life long guide that carried a 500 nitro h&h rifle (an actual Holland and Holland rifle, aaaah!!) for years, he said when you shot a bear with it, it was like "dropping a crane" on that bear!
my 45/70 lever gun is stainless/laminate wood, shot several bears with it using 350gr bullets at 2150 fps. once in a 12 yard shoot out. that rifle was always a definitive stopper. hard to image a gun that can put more firepower on target at a close range. that said, it was at its best at 200 yards or less.
as noted above, some shots can range to 250 yards (which i would likely not allow my client to shoot at, but depends on the cover/terrain) bolt rifles just do that chore BETTER.
so, as a backup rifle, a double would do a fine job, best at ranges under 200 yards. and waaay better under 100 yards. my 450/400 shoots 3" low at 200, 12" low at 250 and 24" low at 300 yards. on a moose or caribou in the open, with a rest, that is doable. BUT, on a bear near the alders shooting beyond 200 would be irresponsible with my double.
not trying to leave black bears out, but they are a completely different thing. a 270 win (i prefer 7mm as a starting point) and up does a fine job on black bears. its not that they cannot be dangerous, but mostly are not.
yes, years ago and in the villages today, hunters shot bears with rifles that are a bit lighter in caliber than i would PREFER. as stated, bears are targets of opportunity. but much like a big 5 safari, a guided brown bear is a fairly expensive hunting experience so a "PROPER" rifle should be used as it will be one of the least expensive part of the whole endeavor.
so in summary, double rifles can be used, but are a bit fragile for most coastal hunts. they are accurate enough (my double wears a 1-6X scope) if ranges are kept within the hunters abilities. with open sights, there will be opportunities lost due to distance. But for nearly everyone, a scope sighted bolt action rifle in a medium caliber (or larger) is an excellent choice for hunting the large bears.
i hope this filled in some blanks for you sir and of course my experiences are not all encompassing or absolute. these are just thoughts from watching and hunting these amazing animals for nearly 4 decades. (damn, i starting to get old!!)
Don