How would you handle this?
There are many possible rules you might be breaking. 500ft from a dwelling, 500 ft from a public highway, discharging a firearm within (city/village/etc) limits, "hunting" out of season, or without a license, and probably some others that might apply.
It really all depends on what the specific laws and ordnances of your locality are.
And while you can call the cops for coyotes, they usually won't do anything about them. Its not their job. The best the police/sheriff will do is call animal control. The least they will do is tell you to call Animal control.
Now, if its an animal vs a human, call the cops AND call animal control. Hopefully someone will respond before its too late. If you are going to respond, I urge you to get legal advice and think it through before such a situation can arise.
So you neighbor is anti-gun. Too bad for them. You might talk to them and mention what happened with their dog and the coyotes. Don't mention your AR, but if it seems agreeable, mention that you have a gun, and would they want you to protect their dog, if you could?
Once you get their reaction to that, you know where you stand.
BTW, does your neighbor have any kids? They might want to think about that, if they do...
Wild mammals (that attack) are going to be considered rabid unless it can be proven otherwise. TO prove it, you need the brain of the animal to examine in a lab. If you cannot prove it (animal escapes - wounded or otherwise), or you destroy the brain because you are such a fine shot, then a person who was bitten will be getting the Rabies shots. A pet,...I don't know, maybe the shots, maybe just put down.
How would I handle it? I'm in a different place. Miles from town, and more than a few hundred yards to the nearest neighbor. Out here, when it comes to coyotes (or other dangerous predators) most folks follow the Triple S rule.
Shoot 'em, Shovel em, and Shut up about it...