If you are trying to save furs, use a .223. Even in open country, if you are calling them in, the vast majority will come to well within range. It's a little much for coons and bobcats, but it works great on coyotes. I would assume it would also work on cougar, but I've never shot one.

A .243 is a little much for coyotes if you are saving furs, unless you are shooting coyotes at long ranges in open country. If you are calling them in, odds are most of your shots will be close enough for a .223. A .243 is way too much for coons and bobcats. Probably works great on cougars.