I might give it a shot. After all, if we were just normal travelers , and didn't bother checking other booking channels and the other airlines, what would they do?

I'd venture to say that if you book it, give it a day, and then call and tell them that you called BA to adjust something in the PNR (lets say seats), and were told that the flight doesn't exist, they may have to re-book you. Any confirmations here from someone more experienced? I don't think this is fraud, but merely taking advantage of a slip-up by the company's IT department.