Sir
I have read this thread all the way thru, and have quoted your queries and concerns. I really like .22lr bird shot, for fun and critter control. Here is my take, for what it is worth.
Most .22lr pistols will not cycle shotshells. The rifling in the barrel WILL cause the shot to open up very fast, making effective range severely short. And in short barrels, shotshells are not exactly quiet. I have a hand-me-down Stevens single shot, with a worn out barrel, not much rifling left, that I use as a garden gun. With the ~14" barrel, it is quiet enough to not bother anyone. You could try for that, although it would be awkward to carry around. If you search for garden guns, which are smooth bore rifle caliber firearms, you can find some that are affordable. Henry USA is the only company making a current garden gun, nice .22lr lever action.
I have also used Aquila Super Colibri to good effect on critter removal. Within limits, the 20gr slug can take out nuisance squirrels, and put down raccoons. They are a primer only powered .22lr round, and out of a handgun are very quiet, and effective.
So, after all this, my recommendation is a small, snub nose style .22lr revolver, loaded with Super Colibris, and maybe one or two shot shells. That combo should dispatch and slithering varmint coiled up in the pasture. I can even recommend a North American Arms mini revolver, in .22lr. The Colibis would still be quiet, tho the shot shells would be a bit louder. 5 rounds, single action, shirt pocket carry sized tool.
Hope you are happy with whichever handgun you get, and that it meets your needs.
Thanks may be time to add a revolver.