Sorry about your dog as well!
I read the whole thread and concur that toxicants are not wise save for one (lead poisoning).
I cattle ranch out west and deal with coyote, cougar, bear and soon wolves. We also have tons of trouble with elk in our hay stacks, hay fields and even grazing lands. I do lethal and non lethal and hazing depending on the species and the location.
Several things stand out in your case. First is neighborhood cooperation, keep that. Toxic substances will lose that, so will fish hooks. What you need to do is meet mr. coyote on YOUR terms, not his. Some trail cams that have quick trigger times and are black ops are important. Placing a cam looking down a trail rather than across it is important. Another thing to look at is some detergents. A unit called Predator eyes might help to first spook the coyote off then later to push it to your satisfaction into where you gain an advantage.
A livestock operator was getting hammered in SW Oregon recently by wolves. He installed a couple of air dancers around his farm and he gets to sleep in his bed now (for the time being).
Now as much as your neighbors wanting this coyote dead the addition of live treats bringing all sorts of predators in much complicates your situation. There is a saying that if WWIII happens that coyotes and cockroaches will surely survive applies. You can kill this coyote deader than dead and 50 will take up it's place. It might be six days or six months but sooner or later, you will have another one wanting to do "ethnic cleansing".
As you have gotten excited at your first attempt at killing this coyote that killed your dog a shotgun might be a good choice. Otherwise, your rifle options are sufficient. It goes without saying that one thing hunters are good at, recommending the purchase of another gun, but your fish you need to fry are in the strategy department and elsewhere in my opinion.
Traps and snares.
They are formidable tools for sure. However there is a learning curve and your neighborhood is likely to fracture their support in the likelihood you catch a non target critter. Or the law will be descending on you.
Best regards and the best of luck.
Three44s