I wouldnt shoot the cop.
But if some bitch shot my dog, I would get her name & badge #. Then, using her name, go online and find tax records, property records, criminal records or whatever and find out where she lived. I think you could come up with your own ideas from there.
You could also spread news of the incident to every available news force and paint her as cruel animal killer (which she is, but you could exaggerate the effort).
Or, hire a PI to follow the cop around when she's off duty and see if you can catch her engaged in any "conduct unbecoming of an officer".
Make sure to file complaints with every office you can, his superiors, etc. and encourage your friends to do it also. Dig through her trash and see what you can find.
The bottom line is that when you're retaliating against someone in that position, you cant be blatant or direct. You have to be every bit as dirty and scheming as the po would be. If you cut off the head of the officers dog and threw it through her window 2 days after she shot your dog, they'd probably have a pretty good idea who did it.