My experience calling in predators is restricted to coyotes and bobcats so what I'll say is supposition on my part
FWIW.usually, either a coyote of bobcat, if you miss them after they come into a call will never come back to either a call or the real thing again.
There is that old saying, "Smart like a fox." so just possibly he may not come back to the call. You never know.
I agree with the choice of the shotgun and #4 buck for several reasons. One, you might only get a running shot and two, there's the possibility of ricochets with the .22, and that's bad news.
I'll grant you the shotgun is a lot louder, but if you only fire one shot, odds are they'll never figure out where it came from, especially if you can shoot from a bit inside a doorway or other enclosure. Even if you're put in the open, it's hard to figure out where the shot came from.
Oh! If you do get him, wear plastic gloves when disposing of the body just in cast it might be rabid. Make sure you don't come in contact with any of the saliva. I hear the shots for rabies hurt like hell.
Good luck.
Paul B.