It depends on what kind of varminting you want to do. If your rifle will be dedicated to going after coyotes, beavers, and crows as you mentioned, then go with the 22-250. You will likely not shoot more than 20 rounds on a good day.

If you plan on doing any high volume shooting like prarie dogs or western ground squirrels, consider the .223. Brass is cheap, barrels heat slower, and you will be hard pressed to see meaningful performance differences under 250 yards. You can also stuff a whole lot more of them in your pockets when leaving the truck behind.