7mag simply doesnt' have enough energy at 1000yds.
For your situation I'd say the smallest you can get away with would be 300wm, and 1000yds on elk even with that is a real stretch.
The safer, and more reliable bet would be 7mm STW, 7mm RUM, or 300 Rum, in a bolt gun.
You should also consider an AR in .325wsm. The AR operating system alone reduces felt recoil considerably, so will the "tube gun" design as it brings the recoil more straight back without the tendency of muzzle flip being as pronounced.
A long 26-28" bull barrel in an AR platform would probably give you the best combination of tolerable recoil and long range power.
If you go with a bolt gun here's two things. Get a heavier Varmint/Tactical/Sniper type stock. In the rear place a 16oz mercury recoil dampener, and just forward of the lug you can place one under the barrel in the foregrip of the stock as well; though you may need a slightly smaller one.
When you have the front one set properly with full free float etc, you can fill the rest of the void with a mix of epoxy and lead shot.
Then of course get a first rate muzzle brake
, and good magnum recoil pad such as the limbsavers to round out the package.
You could easily end up with a 14-18lbs rig in the end but if you dont' mind a heavy rig this would be the best way to achieve your ends with a bolt gun.
Heavy is a PITA on long hikes, but heavier is steadier and helps greatly to reduce felt recoil.