Welcome JD, pull up a log. Nice to hear you rediscovered the Contender, they do become a disease that is hard to shake (took almost 43 years for poor health and handicaps to cure my Contenderitis). There are lots of knowledgeable folks here that will gladly help you with any questions.
There are lots of used Contender era barrels always for sale here, and almost everywhere. A board sponsor, Ed's TC's is another good source and he's a great gent to deal with. As already said, almost all the G2 Contender era barrels will work on your Contender frame as well.
Caliber choice for predator hunting is endless if you reload and/or wildcat, but there are plenty of very serviceable factory barrel choices as well. What predators at what ranges, shooting handgun or carbine, and whether you save the pelts will be your guide. They'll all kill them if you are close enough to place your shot, some even at surprising ranges from short barrels. Some are pelt friendly, some are not. If you don't reload, of the three you mentioned the 223 will be the least expensive to buy commercial ammo, the 22 Hornet the most. If you do reload the 204 is an excellent choice, not hard on pelts and flatter shooting than the other two if at longer ranges. If you reload and buying a custom barrel is an option, you have at least 240 cartridge choices to pick from for your frame, and many of the wildcats are very easy to do. My favorite predator barrels, whether handgun, carbine or rifle were the fur friendly 17's, most of them wildcats (except the 17HMR and 17 Remington). The 17M4 has been commercialized as the 17 Fireball now and it would be an excellent choice without reloading or wildcatting in handgun length barrels.