A long barreled 300RUM launching 215 to 240gr MK or VLD bullets will easily hammer game at those ranges. My rig (35" barrel) gets the 240gr MK to 3075fps and at 1000m, it almost shoots through my steel gongs.
This is not a portable gun at 35lbs. You also need the bench to have any semblance of accuracy at those ranges.
If you decide to go 338, go with the 338 Mystic/Tomahawk or the Wolf. All of these are improved cases based on the RUM or Lapua. With the 300gr MK, you need all the case capacity and barrel length to get out to 1500yds. This would hit significantly harder then the 240gr 30cal.
Jerry