My personal preference would be the .243win, but either will treat you just fine. If I were picking based on maximum performance, the .25-06 would be the pick, same velocity with ~20% higher bullet weights, or higher velocity with equivalent weights. But the advantage isn't huge in the field, as most fellas aren't able to capitalize on that advantage in hunting fields. If you can field shoot 1,000yrds, then the 25-06 pulls away, but at the normal 600yrds or less MAXIMUM ranges guys hunt, either will knock coyotes down clean.

EDIT: You could save yourself some money and just use that AR. The 223rem is my go-to, almost always in an AR-15. For super long coyote poking, the 243WSSM or Win are great. For shooting dogs farther than I really should, a 284win has been my go to lately (was formerly a 6.5-284).