I have a .223, .22-250, and .243 and shoot coyotes, p-dogs, and deer.
1) For p-dogs, I'd MUCH rather shoot a .223 than either of your other choices. I have the other calibers, I shoot the .223.
2) Are you saving fur on the coyotes? Then I'd go with another .223 in bolt-action. However, what are you going to do- carry both and use the bolt-action for longer shots? In reality, you're going to have to make a decision as to which gun you'll use that day when coyote hunting. If you're not saving fur, then a .243 is the way to go. However, if you're not saving fur, then why not just use your deer rifle? No better way to really get to know it.
3) With a .223 and a deer caliber (.280 in my case), I almost never shoot either my .22-250 or .243. I'd sell the .22-250 except that it was a gift (before I got the .223) and I gave the .243 to my daughter- it's now her deer rifle.
Bottom line- if it were me, I'd get a .223 in bolt action and use it mostly for prairie dogs, with secondary use as a coyote gun, keeping the AR as the primary 'yote gun.