The .204 and .223 is too much for all you listed except coyotes. The .17 rem I've blown up too many fox with but great for coyotes as well. .17 Fireball is about the best compromise the is. .17 Hornet is perfect for Fox,coon,and cats. If I was you I would load TSX bullets for the .243 you have for coyotes. Would work for fur. Don't know if you reload or not, I imagine they have factory loads for .243 loaded with TSX. Just please don't use FMJ bullets. You will wound more than you can recover. The TSX will work for coyotes and deer with the .243. That leaves you with the .22 Hornet for fox, coon, and cats if you want a factory loading. The Hornet will work on coyotes if called in close. Within 150 Yards. Stay off the larger bones like the shoulder and you'll be fine. Do not use the V-max out of the Hornet for coyotes.
I don't know what area of the country you hunt in, but if you call you can expect close fast action. Where I am I mostly call grey fox in thick tight cover. Shots are usually within 50 yards. That makes a 12Ga #2 shot the best choice for fox and use the smaller buck shot on coyotes, and the Hornet second choice. If I'm night calling for fox the 12Ga is #1. If I'm night calling for whatever (fox, coyotes, coons) then the Hornet is #1. 10ga is great for fox and coyotes. Don't use a shotgun on coons if your keeping the fur. Fox and coyotes are sold fur out where coon is skin out. All those pellet holes in the coon will fetch a lower price. Don't know if your looking to sell fur or what. If your looking to sell then specialized rifles and tactics are needed. Like a .17FB or .17 hornet for when your targeting Fox, and cats. .17Rem for coyotes. And a .17HMR or .22 mag for coons.
5mm Rim fire is a good compromise too, don't know how destructive the new ammo is for it though.
Many of us have tried and failed to find 1 all round fur rifle for predators. Fact of the matter It doesn't exist. There's so many variables. I have learned that you just have to pick the predator you are most likely to encounter and go with the right rifle for that. Theres a big difference between a 55Lb. coyote and a 12 Lb Fox.