Concern yourself more with the rifle and it's sighting system rather than with the cartridge. Any modern, moderate capacity rifle cartridge with a bullet diameter of .257" or greater will work fine if you choose an appropriate bullet. Over thinking the problem will undoubtedly result in a poor choice. What is important is choosing a rifle that is light enough to pack around easily, and that you can shoot with confidence out to 300 yards. This is a simple yet complex requirement, as one needs to think in terms of a general purpose rifle rather than a mission specific rifle. These threads are filled with rifle choices that are good for one thing only, but do that one thing little better than a more general piece. Pick a carbine length bolt action deer rifle chambered in .257, .260 Remington, 7-08, or .308 , mount a good quality variable low power scope like a 1.5-6X, a 2-7X or a 2.5-8X in good mounts, and start shooting. If a 7-08 and a 7mm magnum are both sighted for 250 yards, what advantage does the magnum have out to 300 yards? Can you use the difference in the field? Will the sheep be able to tell which bullet struck him? The dubious advantage of the magnum is at the cost of a longer, heavier, bulkier rifle, heavier ammunition, and more recoil.
Field marksmanship is more critical than is the choice of cartridge. When you can shoot from field positions and make a first round hit on an 8" aiming point at unknown ranges out to 300 yards, and be able to do it on demand, regardless of wind, or target angle, you will be ready for any North American big game.