Actually, they already sold the seat... to you.

Seriously, though: Unless you have a screen capture or print out of the last step in the booking process clearly stating the correct date, it's unlikely that you can convince BA that it was their fault. There's an automated process that runs thousands of times each day on one side, and human interaction late at night on a website that tends to reset form fields upon changing unrelated fields (I'm talking about you, Flight class field) on the other side.

If you are convinced that you had the right date right to the end of the process, I'd take this up with the credit card company instead.