Glamdring: I'm in accord with your comment about less-than-elk. (Now, if you get into the "pipsqueak" guns of less than, roughly, .30-'06, bullet styles become somewhat important.)
Weasel words: Based on posts at TFL from folks who say they've hunted Africa
, the .375 H&H seems like a good "compromise" cartridge. (I'll stay away from "what bullet", cause I ain't dunnit.)
It's flatter shooting than the .416 stuff, so it oughta work in your mountain hunting. People keep saying that it works on Mr. Buffalo in Africa, with appropriate solids. Capstick, et al, have approved of it for lions and leopards.
And that's a capsule summary of decades of reading and dreaming.
, Art