I've shot 2 coyotes 1 with a bow , a 48 lb recurve with a 125 gr Bear Magnum total wt 527 grs,passed through at 20 yrds that 25 lb dog ran a mile to get hit by a Mack. The 2nd 1 I shot with a 12ga from my duck blind ,30yrds 1 steel dead dog in 50yrds ruined the decoy too. Funny thing the shotgun dog had just 1 bb thru-through the ribs maybe 6 under the skin. The bow dog based on equiptment shoulda been dead on the spot but ran a long way and would of gone further. The shotgun dog was just unlucky as hell.
While I agree that a 223 is about as light as you should go I can see no reason why used responsibly taking best shots at short range woould be any worse than the glorified turkey loads suggested by others here.