The 22LR will kill Coyotes,,,, problem is,, 99.9% of the time they'll run off to die a slow death..
I applaud the fact that you want to save the fur but show the prey some respect,,, use the 243 and learn how to sew..
If ya don't reload,, there's some 55 grain factory loads out there that can't be any worse than the 55gr 22/250 stuff some guys swear by..
I'd use Soft Points, or Hollow Points rather than the trendy plastic tip stuff,, but that's just me..
Good Luck
Charlie