Yes I reload and have an AR15 with a 20" RRA heavy varmint barrel and have a FN SPRA1 in .308. The purpose of this build is for prairie dog out to 300yds.
I'd say that ammunition availability/pricing would be a major factor in my decision making progress.
Personally I tend to stick with the most common caliber / equipment types I can get when adding firearms. If I were in your situation I'd be looking at .223 simply because there are tons of parts/ammunition for a rem700 in .223.
If you're worried about ballistics performance for longer ranges I'd be looking at moving up to something like .308 instead of a higher performance/more obscure .22 platform.
Gotta define the mission first. Varmints at close/long range? What size? What level of accuracy do you need?
The .223 will take care of most varmits out to a couple hundred yards with performance falling off as the distance/size of varmint increases. It's also the cheapest to shoot and easiest to shoot.
Shooting medium range in heavy winds and the .243 is a better bet. Barrel life is intermediate.
The .22-250 will flat shoot out to 400 yards pretty well and is good for small targets at variable ranges (less need to figure bullet drop), but burns barrels faster than any of the three.