None of those are wickedly notorious barrel burners. I like the .25-06 better on deer. I have a friend who hunts coyotes for a living in the winter. He uses a .22-250, but has cleanly taken a dozen or so deer with it.
CZ...or maybe it was Baikal, makes a nice over/under in .243/12ga or 20ga. This might be okay if you aren't shooting sod poodles at 500 yards.
Big All,
I agree, it would be a PITA. Cottontails here are not flighty, so it is doable but yes, a PITA. I only mentioned it to stick to the spirit of the original post. A second gun, even a beat up $50 H&R topper or better yet the $100 .22 you suggested would be FAR more appropriate.