Welcome Bobcat 46,
I am also new to the forum here and am trying to call coyotes and bobcats out west (Colorado). All I can help you with is the taxidermy end of your quest. I will tell you that I have seen some coyotes unusable after being shot with a .243 as the exit hole took out one side. Bobcats with slightly thinner skin may do more damage. A 22-250 does pretty good damage but didn't seem to ruin them. I shot a coyote once with a 90 grain FMJ out of a .243 but the coyote ran the length of the field before it dropped and had I been in the brush, would have lost him. This would probably work ok with snow on the ground for tracking. I just purchased a .223 for bobcat and coyote hunting but can't comment on it just yet, haven't been lucky yet. I called in a coyote yesterday and it came from behind me. Must have done something wrong, had the wind to my back and the coyote trotted right on by 20 yrds away and kept on going. A load of buckshot would have been nice to have at the time, but never offered a rifle shot.
On the bobcat, if you want one mounted, freeze them whole. Some taxidermist skin them differently than others. We make a single cut down the abdomen, and other make dorsal cuts, so I wouldn't skin them, let the taxidermist do this. You can see some of our work by visiting the sit in the above post. Good luck with the hunt and I'll post the .223 damage later.