Pistol? You Betcha!
Coyotes to 500 yards with specialty handguns is not to much of a problem as far as hitting them. The key of course is getting the bullet to perform the way you want at varying distances.
I would opt for a bolt pistol before a break-open but that is preference on my part.
You are going to shoot approximately 200-300 fps slower in a specialty handgun than you do in a rifle.
If you want 243 Win performance consider a 6-284.
If you want 6.5-284 performance go with a 6.5WSM.
My hunting buddy Steve Hugel (sscoyote) longest shot on a yote was just over 900 yards. He has made a # of other kills in the 350-600 yard range with pistols.
In SP's he has a 17 Mach IV, 20BR, (soon to be)6.5WSM, and 7mmx270WSM. These are not just for coyote but PD's and big game.
My rigs are: 22-250AI, 260 Rem, 250 Sav.AI.
My main LR rigs are: 6mm-284, several 6.5-284's, and a soon to arrive a 7T Lilja 7mm Dakota.
The 6-284, two of the 6.5-284's and the Dakota are XP-100's (1 rear-grip 6.5-284 and all the rest are center-grip).