I've decided to build a rifle designed for Western elk that is capable with long range shots. What do you think is the ideal cartridge for this purpose? So far I've shortlisted:
First, what do you consider long range?
None of the cartridges you list are really truly long range cartridges.
The 8mm is the closest with ballistics very similar to the 300 Win Mag. The 338 Winchester offers little in improvement over the 30-06 as far as long range ballistics, it does however produce more energy with bullets of a little better sectional density, increased bullet weight, and increases frontal area.
The 350 Remington Magnum is not a long range cartridge and offers no improvement over the 35 Whelen. The 375 H&H magnum is not really a long range cartridge however it can be made to work in a pinch.
If recoil is not a problem, and I assume it is not since you are considering the 375 H&H magnum, then I would consider the 300 RUM, 300 Weatherby, 30-378 Weatherby, the 338 RUM, the 340 Weatherby Magnum, 338-378 Weatherby and the 358 STW. These cartridges also produce very significant recoil and a lot of muzzle blast. The Dakota and Lazeroni cartridges should be considered if cost is not an option.
If one is a little more recoil sensitive then consider the 7mm Remington Magnum, 7 STW, and 300 Winchester Magnum.
Honestly any of those mentioned cartridges produce adequate energy with bullets of adequate weight and sectional density to kill elk beyond 400 yards.
Offers no ballistic improvement over the 35 Whelen and is not really a long range cartridge.
Good luck.