The .223rem is the most practical and suitable cartridge for that purpose, assuming you care about recovering hides. The .22-250 and .243win can get a bit messy on badgers and coyotes, and the .204ruger gives up a bit of range for its super-light bullets. The .223rem sits right on the sweet spot with 50-55grn bullets for coyotes and badger. Affordable and readily available ammo, minimal reloading cost, and plenty of performance, the 223rem is where it's at.

If you don't care about recovery of pelts, then the .243win is the most practical. It'll get messy on badgers especially, and often on coyotes, but it'll anchor dogs with authority way, way downtown.

If you're looking for a superb rifle and if you're interested in a heavy barreled varmint rifle, then the Savage 12 is a fantastic option. The BVSS Varminter is my favored stock model, but the newer VLP-DM Low Profile Varminter (detachable box mag) is also a fantastic stock. My wife's 12bvss in 223 was under 0.2moa out of the box, my 223 was under 0.6moa and my 22-250 version was under .4moa.

For budget friendly models, a Remington 700 ADL can be had for under $400. The Savage Trophy Hunter 11 is a good option in this price range as well. I bought a .22-250 Trophy Hunter for my dad two years ago that printed under 1moa, and a recent ADL 243win that I bought was under 0.8moa after bedding and lapping the lugs.