Keith, I was replying to the OP, not you if that's how you read it. I've known you long enough both on and off the forum to know you are an ethical hunter.
Those of us with lots of P&V and LR experience know that picking the cartridge and load is just one page in a book full of esential requirements to drop a coyote where it stands at longer ranges, and most of the rest of the pages are equally if not more important than that one is.
For starters it wouldn't be with a 223 for 500 yards, and unless the circumstances were just right I'd even think twice about either the 204 or 22-250 (which in general terms are both superior to the 223 ballistically and with 40 gr & 50 gr bullets respectively can be loaded pretty close to ballistically the same at 500 yards). Sure all of them have killed coyotes where they stand, and some even at 500 yards, but they are not the ethical choice and put greater demands on the abilities of the shooter. To kill at that range reliably you need to assess the range accurately, know your firearm and load intimately, then aim and hit small with a bullet retaining enough energy to get the job done. IMO not all that many predator hunters are capable of making good hits at long range from field postions, so "more" is better for them.
From the OP's list the 243 is probably the best choice, but there are others that would be even better choices than it.
