We used to do the coastal hunt for a living before the BC grizzly closure, and I was totally happy with a client opting for .270 / 7 Mag / .30-06 / .300 / .338 / .375 Mag. .270 is the light end but no denying it will work. 7x57 175gr is the lightest round I’ve taken a grizzly with (brown bear to Americans when coastal, same species) and while perfectly lethal it was the least dramatic kill on a heart / lung shot, and it had a long death walk. Not a big deal on a moose, quite a big deal on a grizzly bear you have to follow a hundred or a hundred and fifty yards into the nasty thick stuff they choose to die in. Those touting Corbett and Bell need to remember Bell made head shots exclusively with FMJs on Elephant, and that Tigers are smaller than Grizzlies and Elk and more lightly built.
If I had to choose the ideal coastal grizz / Brown round .300 for the hunter, .375 for the guide. I’ve seen no discernible difference in effectiveness for a hunting shot between the .300 and .375 on Grizzly / brown, but do believe in the .375’s greater ability on bone in a frontal / charge shot or to break a shoulder of a bear running somewhere you don’t want it to go. If I had to choose the lightest round I’d tackle the NA 29 animals with, yea it’d be the .270, and I’d do it happily. However the .270 is illegal in BC for wild bison, so that’s out.
Then since we’re talking the North America 29 big game animals, we have to recognise the elephant in the room; mountain game. For all the noble talk of keeping it sporting and fair to the animal with a 12 gauge, .45-70, .348 Winchester or the like the reality of it to those who know the hunt is you’ll need to be able to make a 400 yard shot one day. Mountain goats in particular will drive this point home, all the comments from East and central Canada seem to be understandably out of touch with the realities of hunting above the tree line on open ice and rock. There’s utterly nothing more sporting than a stalk up terrain that will kill you with a misstep, that takes two full days of all our exertion to get to 375 yards (which looks like 100 yards on a deer in the bush, over open rock) on the billy or ram of the hunter’s dreams.
All this adds up to 7 Mag for me, for the OP’s question.
You need a 175 grain bullet with greater than 2000 ft-lbs of energy at 100 meters for these guys that can weigh what three eastern moose do; check for the 7 Mag.
You need to be able to make a 450 yard shot cleanly on picture 3 & 4 over the terrain birds eyes in pics 5 & 6; check for 7 Mag at 140 grain. The challenge of this hunt isn’t the stalk, it’s surviving and getting to the shot.
You need to be able to anchor this guy at 25 - 250 yards; check for 7 Mag.
AND all this needs to be done with the OP’s stipulation of the minimum possible recoil for the maximum capability; check for 7 Mag, which is extremely similar to a .30-06’s recoil. A .300 will do slightly better at most of this but will recoil more, even without the OP’s criteria I personally like the big 7. If people are basing their opinion on their experience hunting moose, black bear, and deer in regular bush they should give thought to what the breath of this continent holds beyond deer and moose hunting.