You can either shoot what you already own or use this as an excuse to buy something new.

For pronghorn, it would be hard to beat a .25-06 or a .257 Weatherby., but lots of cartridges will work, from the diminutive .243 to the demonstrative .338 Win Mag (probably no more meat damage than a .243 or .25-06, as the .338 Win Mag will probably pass through with no expansion). Or use your .338 Lapua -- same idea, bullet won't expand so probably won't damage meat much -- just drill a .34" hole right through the pronghorn. If it is as accurate as you say it is, you can skip directly to "ready-to-go" with a load that works well in your rifle.

Generally you don't need to shoot 300-400 yards.