BrianK,
As a very young kid (8-12 y.o.) I use to follow my grandfather on his trapline. Where we lived in Northwest MN we didn’t have any coyotes, but we had a lot of fox. He also trapped coon, beaver and mink.
He used “pan traps” made by Blake and Lamb mostly, but did use snares as well. Every year he would catch someones poor farm dog. This was an area where people were spread out several miles apart.
I would not recommend snaring or trapping as close to a residential area as you seem to be.
Would a shotgun work? Rifle seems dangerous, but if it’s legal, and you know the area you are shooting in, I would think you’re alright. Maybe keep your caliber small and only take the close shots.
I personally have never “hunted” predators, but have always wanted to. Seems like it would be fun and very challenging.
Scott