Ok, just read the thread again. I didn't read anywhere here that the OP (@James.K.Polk) was getting this repaired anywhere other than by Apple, but getting the insurance from the CC provider to pay for it once it's repaired. (so the initial charge would be on the same card).The OP is not getting Apple to fix it but via some insurance. When the OP goes online to register with Apple Care + the original details and serial number of the mac will be used. The OP sent the mac to Apple support, they did some checking and diagnosis on the unit. The unit had no warranty cover because the OP admits it was something that was overlooked at the time. Apple quoted an out of warranty repair cost which the OP cannot afford hence using other insurance the OP has.
Now due to having sent the unit to Apple, they will have the unit's serial number on file so the question remains will online registration of Apple Care + flag it up as cannot be used because Apple will have no way of knowing that the machine that was originally sent in as faulty, has been repaired. As you rightly pointed out in your first post about the issue of fraud, Apple would need to make sure that the OP is not signing up to Apple Care + with the intention of making a claim soon after being registered.
Like i said, the OP will have to live with the doubts that Apple may have about the legitimacy of any Apple Care + registrations that are made.
If the repair is done by Apple, then there's no question. (and I don't know why on earth you would do anything but).