Well, I'd recommend the 223 first, 22-250 second, and 243 third. Each has their advantages and disadvantages, but for what you're going after, coyotes and jacks, the 223 has plenty of punch and very diverse ammo offerings, and 223 ammo is cheap, brass is cheap, etc. The 22-250 and 243 have longer range capabilities, but unless you're shooting past 300 yards regularly the 22-250 and 243 are probably more gun than you need, and are more expensive to shoot.
I'd also check out both the bull and standard barrel to see which you like better. The standard barrel is still plenty thick and the balance and weight may be more to your liking. Other like the balance and weight of the bull barrel. So I'd see which you like better then buy it and a few boxes of shells and go to it!
later,
scruffy
