Success! A very nice lady at First USA just called me and told me that the credits would be added to my new Platinum card. (I had figured that having the Platinum card was not only a good idea in its own right, but would allow First USA to give me the credits without reprogramming their computers.)

The moral of the story is that you have to go high enough in the organization to get someone who is allowed to use his or her brain power and common sense. I figured that Southwest's Customer Relations people would have the appropriate contacts, and they did. First USA responded immediately, and showed genuine concern toward fixing the problems with their system. They have earned my loyalty.