Are you stuck on bolt action? Cuz this is the first thing that comes to my mind when I think “backpacking through the back country”
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As others have posted I would recommend stainless, based on you usage of "this rifle will be subject to rain and snow for days on end" it would be worthwhile .
For brand recommendations, you'll find this limited with open sights, so you'll have to choose between the few available, I'd say try shouldering whatever is local and then buy what feels nicest in your hands, but you'll likely go with a ruger just based on availability, it's not a bad option though either, but my preference would be a tikka or steyr prohunter. Actually a stainless steyr scout would be good to but over your your expected budget.
From the late 60's many many backpack sheep, goat and caribou hunts in the BC mountains, sometimes for a week or ten days straight. In the early years before stainless rifles were available my go to mountain rifle was a Husqvarna M4000 featherweight .270 and I always carried a tiny bottle of oil, an oily rag and a pull through and I was always able to prvent rusting. Put tape over your muzzle and just shoot through it, a waterproof sleeve for the gun works great if not anticipating a quick shot. Inlater years I tried numerous other guns that were usually too heavy and always came back to the Husky. My last mountain rifles were Browning A Bolt Stainless Stalkers and they were true all weather warriors. This is what I would recommend today in a non magnum ( lighter weight) backpack rifle. J
There isn’t much to be gained by putting iron sights on a scopes rifle, as long as you don’t cheap out on the scope.




























