Since you didn't detail, let me ask these questions for you to consider.
how portable do you need this rifle to be? Walk around with it or just out of the truck and good to go?
Will you use a muzzle brake
How much hunting will you REALLY do beyond 750yds? Be honest, this is your hunting spot. Only you know the answer to this.
How windy is the valley?
Will you need a quick follow up shot?
How much of your budget are you willing to invest in support optics? This stuff ain't cheap, well could be...
There is little doubt that the 338 in just about any cartridge is a boomer. By answering the above, you will quickly narrow down your choices. I hunt under similar conditions and don't use a 338 because I don't need to.
Ballistically, 250 to 300gr bullets are you best bet. But unless you get them over 2800fps, at range, wind can buck them around. In a mountain valley, I suspect updrafts and crosswinds are common and sudden. So now we are talking faster like 3000 to 3200fps. That's a very big cannon.
Consider that all you really need is a heavy for cal bullet from 7mm to 338 to do this job.
Personally, with wind being a constant enemy, I would go for the fastest cartridge to launch a suitable bullet. Any bullet in the 162/175gr and up arriving at 2200fps and faster is plenty lethal for your intended target and range.
I would look at a fast 7mm with a hot RM as a min, better a STW or a RUM if you can handle a 28"+ barrel. Launch 162gr AMax as a min, 175 to 180 better, some of Wildcat super heavies as an insurance policy.
Wind drift will be less then a slow 338 or a fast 225/250gr bullet. Impact energy is more then sufficient (my loads arrive at 1000yds with over 1800ft/lbs but more importantly, over 2400fps).
Next up, the 30 cal with heavies. 300WM is a marginal min. 300Wby and 200gr accubonds is nice. A RUM going faster is nicer (220gr MK or 210gr Bergers would do the job nicely). Recoil at this level without a muzzle brake or a very heavy rifle is stout. Certainly way to much for you to spot your own hits.
Are you hunting with a spotter?
The 338 are boomers but create their own series of problem. I don't like LR hunting with any rig that will not allow me to spot my hits at distance. With or without a spotter, being able to see the hit confirms what I do next way quicker.
When I hunt, the rig has to also function in the way I am going to move about or not. 36" barreled 35lbs cannons don't make for portable rigs. Gentle to shoot monster bullets.
Since it sounds like a single shot is not out of the question, I would keep the action you have and rebarrel. Odds are a standard length fat boy like the Dakota family will work just fine if you do need a repeater. Again, a 7 or 30cal are ideal.
Inside 1000yds, a cannon is nice but unless you can go fast enough wind will limit your range.