Saving fur and the 243 really shouldn't be in the same sentence, it is a crapshoot(sorry) that you won't get a large exit or blow out if near the edges. Shooting them the long way with 55s, 58s and 60s seems to work. Centered 55 gr NBT Lead Frees have worked for me 62gr VGs left huge exits.
I'd just pick a accurate flat shooting bullet to kill coyotes and hope you get a few decent pelts. Sewing works for the ugly ones.
Good luck, good hunting
I've used all the bullets mentioned in my 6x45, 6mm-204 and 243. I abandoned the 243 as I found it to be too much gun for coyote pelt hunting. Even my smaller 6s don't go out when pelts are wanted. I am going to shoot the 75gr Speer HP out of my 6mm ARC this week for coyotes.
Just another thought. Scopes if your going through upgrade from your 3-9 I'd look for something with more FOV on the bottom end and more Xs on the top, Something like a 2-12, faster target acquisition on the bottom end for those ones that appear out of nowhere and a little more power for ones that want to hang up. Bet you'll find the bottom end much more useful than the top.
Boattails vs Flat-base, for a new loader flat-base bullets can be problematic. If you use them be sure to chamfer the inside of the neck, dies like Hornady seaters with a sliding sleave help keep the bullet aligned with the case mouth, when seating gently feels your way so you don't collapse a neck with a cocked bullet. Most of the bullets I shoot are flat-based, I like them.