I have a 17 HMR and have been disappointed in the performance with anything other than a head shot. Head shot, DRT, body shot, run away and die but it is always a long run. Most times I follow the birds after a day or two. Most of my shots are less than a hundred and the longest is 168 yards.
I would opt for the 17 WSM and very good optics. Take the head shot when possible and anything else hope that the 17 WSM doe a better job than the 17 HMR.
Non hunting urban types will NEVER understand the what and why of predator control. Noise, oh that poor, fill in the blank, is just trying to survive. Until they eat you cat, dog, chickens, goat or sheep.
Feral cross bred dogs/coyotes are very scary and very aggressive. I have no compunction about killing every one that I can.
As a LEO you are between a rock and a very hard place. GOOD LUCK!!!