Firstly, since the OP's list here never included "brown bears", let's leave them out of it. If you have enough money to go on a brown bear hunt, you have enough money to buy a rifle JUST for brown bear hunting.

Mule Deer, Elk, and Moose are the possibilities. Of the three, my own experiences have been that Muleys would most likely present the longest shot, since they often share range with pronghorns, as well as with the elk and moose. Though some outfitters will take you on a sagebrush elk hunt, elk tend most often to be found in the timber. You might find yourself staring across a canyon at a herd, but more often than not, it's more likely to be across a meadow and shots shorter than 300 yards (I've never had to take a shot at an elk beyond 150). Moose are going to be encountered up close most often.

If you KNOW, beyond a shadow of a doubt, that the country you'll hunt is open, unobstructed by timber and terrain features, any of the choices here would be okay, though I'd lean toward the 7mm Mag.

One can still find 7mm Mag ammo in almost any country hardware store out here, where the same cannot be said of the other two (I really have to wonder who'd go on a hunt and forget or run out of ammo, though?). Whether or not one reloads, I'm just not sold on the short-mag craze. So you can use a short-action rifle - "that saves weight." Hmmm, I'd venture to say that if weight were really a concern, most of us could drop ten pounds of it between now and hunting season - and none of it would come from the rifle? From what I've seen of them performance-wise, they're not worth my giving up the long-action, belted mags that I already own - in spite of all the marketing hype and the gun writers.

Contrary to what many carry into the field, western hunting isn't necessarily a sniper's dream. Practice snap-shooting. It's a skill you'll need out here, too. Don't be surprised if you find yourself in alternating stands of thick timber, broken up by an occasional meadow. Ditch the bipod and the 12+ power scopes, I rarely use anything above 4x. If you think you'll do some spotting, bring an actual "spotting scope" - unless you wouldn't mind someone like me pointing my rifle at you to watch what YOU are doing? START GETTING INTO SHAPE NOW - you'll be doing a LOT of walking.