It only took about a month, and around 4-5 hours of my time, but the tire saga seems to have come to a close.
I had returned on Monday (the day after the system was down), and the system was still down. I returned once again, at another time, and the system was up. However, the guy who was working that day swore that despite the assurance from the manager that it was approved, he said it wouldn't be, and once it was denied, it couldn't be approved. The manager wasn't there at the time, so I got his number and planned to follow up with him.
Over the next week I tried calling him many times. He had no voice-mail, and never picked up. I was too busy to do much about it until this morning, when I stopped by once again. One of the original guys who I had dealt with was there, and though he denied remembering me he was able to get the approval, albeit for only 75% of the cost, not 100. Considering that only 2 of the 10/32 of depth of the tire had worn, and that their policy is to replace wear <25% at 100% of the cost, I was annoyed. He directed me to the manager on duty, who was not the one I had dealt with earlier.
After I filled her in with the background and explained the conflict with their contract, she sought the assistance of another manager. He went off with the paperwork to look into it. During that time she was trying to figure out a way to make it work with their computer system, and suggested the possibility of giving me the credit I deserved plus the difference in the form of a gift card. I was willing to accept that solution, and she eventually decided to pursue it. She had a clerk ring up the transaction, but something went wrong and the clerk said I owed them about $50! The manager got frustrated (and I had explained that this was approaching 5 hours of my time on this project), so she told the clerk to give me a complete refund on the new tire purchase. This ending up being better than what their contract requires, as normally the contract only covers the original cost of the tire (vs. the inflation affected cost of a new tire) and excludes taxes and installation fees.