a heated barrel won't be an issue, unless you plan on sitting over a highly populated prairie dog town with only one gun to use. .223 is cheaper to shoot. If you think the majority of your shots on coyote sized game will be less than 300 yards, than the .223 is fine. The 22-250 doesn't have that much more recoil, although seeing the impact can sometimes be a problem, if you concider that to be a problem.
It's a tough choice, either way you're getting a new gun. Sounds like a win win situation to me!