Whoa! You WANT a varmint bullet. A light one. It'll make a tiny entrance hole and completely disintegrate inside, leaving no exit hole. The 243 is probably packing too much power to hope for complete disintegration inside the critter, but a 22-250 with a 55gr vmax is about the most pelt-friendly combination you can have for 'yotes. The 17rem with VMAX bullets is also popular for this reason.
Some folks used to advocate FMJ's for fur friendliness before the ultra-explosive plastic tipped varmint bullets came out, but an FMJ will leave 2 caliber-sized holes in the pelt instead of just one. Plus, a critter that you can anchor will yield infinitely more fur than one that gets away.
Unless you're shooting really small fur-bearing critters, I think you're better off with ultra-fast, lightweight explosive varmint bullets. For best terminal performance, pair a lightweight explosive bullet with a fast twist (which would be the most fur-friendly, but probably not the most accurate. But do you really need 1/4 MOA performance?)
As far as rebarrel options, the 22-250 (or the AI version) is probably your best bet. Uses the same boltface, and shouldn't require any magazine alterations to feed well. The Swift might be a little bit too long, and the .222 family of cartridges will require bolt-face alterations. The BR family might also be a good choice - a 22BR is similar in performance to the 22-250. These may require some feed system tweaks at they're quite a bit shorter. A 22 Cheetah (Mk I or II) or 22-243 would be the ultimate in 22 performance, potentially 5,000+ fps with a 40 grain bullet, but with limited barrel life no doubt.