Up where? Idaho doesn't limit center fires so I'll leave that one alone for the time being. In Wyoming it's generally a min of .25" in Nebraska we allow 22s with at least 900 ft-lbs of energy but .25 for elk, south Dakota allows .243 for elk. Lots of people are gonna scream about .243 for elk, but my family has done it handily for quite awhile. Bullet construction makes a HUGE difference. For your purposes, let's start with the .243 with a Remington, ruger, or savage barrel so that you can use the 100 gr gameking just cuz you can. Then there's a whole SLEW of decent .25s but the most prominent are the .257 Roberts (EXPENSIVE) and .25-06. Basically the 06 adds just a LITTLE weight and velocity to the .243 capabilities. Then we jump to the b e a utiful 6.5s that can take game of all sizes on this continent with proper bullet placement and selection. The 6.5 creedmoor, 6.5x55Swede (which is kinda of crippled by factory loads) and .260 rem. (6.5 Grendel is viable at your ranges, but not ideal for knockdown shock and awe). The .243, .260, and 7-08 are all spawned from the .308 case and thus all recoil will be directly relevant to bullet weight. A .243 with 100 gr bullets will kick similarly to the .260 with 100 gr bullets. The .243 can be loaded with 85-87 gr bullets for deer for minimal recoil. The .260 really cranks the 120-140s out there beautifully and they give you better ballistic coefficient and Sectional Density (wind drift resistance/better trajectory, and better penetration) than the 7-08 will in the same loading. For ANY deer, a 100 gr nosler partition, or 120-130 just about anything (except maybe the sst at close range) will work just fine. Elk ? 140 still gets the job done. Our deer get big enough to be drug out by a horse but they aren't so tough yet that a .243 won't knock em flat from 50-400 yds. The .25-06 is just fine too, but it's right between the .243&.260 and I can't justify the extra powder that has to be burned to make it work.
ETA I love the 7-08 and have nothing against it whatsoever, if you think that's your best choice, that's your call and you'll be fine, but you DO have options.