Something that you could also consider is how far will you be shooting coy dogs.
There are places I hunt in Az. that a .223 would do fine, because most shots are less than 200 yards. Not that you cannot hit them farther with the .223, but it lacks power to put them down for keeps. They are among our toughest game to kill. A wounded one will go for a long ways. While they are predators they deserve to be taken with as little pain as possible. If you are calling and shooting coyotes close, then it should work fine. BUT for my coyote shooting here in the plains of Wy. and Mt. I like the big .22's. a .22-250 or a Swift are among my top picks but I usually use them for dogs under 225 yards or in areas that I need a highly explosive bullet. They will hit a coyote much farther, but the loss of energy and wind make them less than top choices for long shooting. For shooting a longer ranges my pal likes his 6mm, use my .25-06, but when the wind kicks up or snow makes longer shots possible we both put either a 7mm Mag. or .300 in the gun rack in the pick up.