This is a simple problem, and it calls for a simple solution. That simple solution is called a Taurus M941 double-action revolver (get the 5" barrel for increased accuracy), and it holds eight .22Mag (aka .22 WMR) rounds. If weight is a concern, get the Titanium model. The .22 Mag is sufficient for any beastie at 40 yards, especially with a quick follow-up shot or two, and it won't splatter anything except maybe a small mouse. You can also get .22Mag shotshells, and you might want to load a couple in the revolver cylinder--my experience, though, is that shotshells are useless beyond 10
feet anyway.
The .410 is a horrible choice, sorry--it's actually quite difficult to get any kind of accuracy with it, and, as another poster pointed out, past 30 yards you might as well throw a handful of sand.
All the larger calibers come with the disadvantage of heavy guns (to offset the recoil). Stick with the revolver (you can also get a Ruger Single Six (it has interchangeable .22 LR and .22 Mag cylinders), but it's more expensive and single-action only. Likewise .22 Mag ammo is cheap too.
Own Drummer
A 22 won't kill a coyote instantly, but it will raise hell with it. If you don't like the weight of a 357 a 22 or 22 mag will work.
That simple solution is called a Taurus M941 double-action revolver Smith&Wesson used to have a nice gun chambered for 22 Jet, and I believe you also got a cylinder for 22mag with it. I also seem to remember they had a model chambered for .30 carabine / 9mm. The 22 Jet model went out of production years ago, though.