For a grab and go fun gun that has the potential to shoot a lot farther and a lot more accurately than I have capability....
Savage Long Range Hunter in 338 Lapua. Decent price, little heavy-ish, but for long range shooting, would imagine you,ll be more inclined to set up and pick your shots.
Big, loud muzzle brake, again, would imagine probly not a deal breaker for setting up and choosing your shots. Big, loud, wonderfully effective muzzle brake. Soft on the shoulder like your girlfriend smacking you with a feather pillow after you,ve said something slightly innapropriate. But not in an angry way..
Fun. Fun,fun, and fun.Man, just thinkin about that gun gets me droolin just a little bit. Lapua brass, little spendy at first,should last a long time.. Plenty of bullets to choose from.

Of the calibers mentioned, and the calibers considered, I,d want to jump up a notch. If I had better eyes, was a bit better shot than I am, and the terrain to take advantage of it, got a sneakin suspicion I,d be sneakin off to the gun shop to get my hands on one the first chance I got.
And that,d leave a nice little niche for a 300 H&H to fill! Just cuz..

How many rounds do you estimate you might shoot in a day? I ask because recoil could become a consideration. Also, how much moving around will you do during a day of shooting. If you're humping the thing around the mountains all day, gun weight becomes an issue.

LWD

Why not try the 338rum. It is the cheap twin to the Lapua. Read the spec too close to make a diffrenence.

Why do you wish to practice shooting at such long distances on live game? IMO that is all you are doing, throwing hail marys for kicks at the expense of the animal. I see this as an extremly narcissitic practice at best. If your going to kill a critter, dont you think it deserves the dignity of a clean kill as well as you are capable of doing it? And a clean kill that offers the best assurance that the dead animal will actualy be found? Are game animals in your parts so abundant that you can shoot them just for shits and giggles and leave their carcass lying about?