I suppose today's part of the saga was fruitful, if predictable. BA's Twitter rep contact me for some information about my tickets, then promptly ignored all further communication. At 4pm, I grew tired of waiting for his reply and called their reservations line. While I could tell the rep wanted to help, there simply wasn't anything he could do. He transferred me to his supervisor, and I lost my cool. I told him in no uncertain terms that my next action would be to file a chargeback with my bank. He apologized profusely, but insisted there wasn't anything he could do.
After 15 minutes of waiting on hold for my bank, I wrapped up my work day and drove to the nearest Bank of America branch. They were very helpful, and quite amazed at how calm I was, considering how much money I was missing. The money will be back in my account tomorrow.
I wish this was the end of the story. I know that I have set off a series of events that could make the last week look relaxing. Chargebacks aren't fun, and that's why I begged and pleaded with British Airways to just give me a refund. Unfortunately, I had no choice but to protect my financial interests. Nothing will stop me from closing on my beautiful new house.