The .223 Rem will do the job from just about any rifle on that size target at that range. The ammo is cheap and readily available and you can play with it if you're a handloader. A lot of these varmint rounds are pretty exotic and hard to find...not to mention expensive.
I use the .223 for medium sized game (coyotes, coons and possums) at ranges out to 300 yds or more from a Ruger Mini-14 topped with a NcStar 3-9X scope. I really like the Wolf varmint rounds with the red plastic tip. They feed reliably and expand nicely without causing too much pelt damage. They are affordable and the quality control seems pretty consistent. That's always important with an autoloader. Good luck!
I agree with the above post. The 223 rem will be excellent for what you need to do and the ammo is everywhere. 22-250 may be another caliber that fits your situation for you to look at that is easy to find ammo for. It is more powerful and flatter shooting than the 223. Out of all these calibers, we are not talking about a whole hell of a lot of difference one way or the other.
Every year many hunters and outdoorsman and women come out west from the midwest and east coast to hunt the prized mulies and elk. One topic that comes up often is altitude sickness. My advice for flatlanders is to get into the best possible shape. Start months before your hunt, usually really ramping up my cardio around March or April.
I run 5-10 miles 3 times a week, and also go for walks carrying my pack with about 50lbs to simulate what could be on my back. Another useful tip is to drink A...