OK, so when I next logged into the game after entering the 12-letter code into the PlayStation Network, the game prompted me on whether to add FFXIV Realm Reborn to my existing service account (that account clearly identified with its current features and my characters' names) or whether to create a new service account for it. With all the confusion elsewhere, it's nice that at least that page was clearly laid out (once I finally managed to get to it, that is).

So the only tricky part was figuring out that the obvious code on the registration card is in fact a red herring that has nothing to do with registering the game. You just need to find the other code in the right format and know to go to PSN to enter it.