My primary calling rifle is a 22 Hornet. I'd likely be using a 17 of some sorts(NOT an HMR), if I didn't have the same dilemma as you, and need to use the same rifle on coyotes, "and" foxes. A 223 will blow a fox up horribly with any factory ammo, and anything but a down-loaded combination (basically within 22 Hornet velocities) that you come up with yourself is gonna blow'em up too. Honestly, it's even hard trying to find a good balance with the 22 Hornet, but it's at least somewhat easier to do than with all the other centerfires.
The 22 Hornet will positively dump a coyote consistently at normal calling ranges. With a medium, to slightly below maximum load combo using a fairly tough bullet that'll offer just "enough" expansion, you'll kill coyotes just fine, and your foxes will have exits ranging from dime sized(rarely) to quarter sized(acceptable) to tennis ball sized(rarely again) when something goes wrong.
I second the shotgun line of thinking. It's truly the only way you can kill both without hurting the foxes hides.