I'm interested to see how this goes for you, as I had a very similar situation recently, albeit with Aerolineas Argentinas, not BA.

About a year ago I booked domestic flights for myself and a friend totalling about 500USD with my BA Amex. The charge was pending for around a week and then dropped off. I contacted Amex who like yourself said that the airline had not claimed the funds. I then contacted the airline on multiple occasions and they were adamant that I had paid and there was no problem with the booking. They even sent me a receipt and said that I must have missed the charge on my statement (I 100% did not).

In the end I resigned myself to a bit of a battle at check in to pay the original fare, however nothing was asked and both flights went without a hitch and I am 500USD better off. The flights were in August and my card has now expired so it will be impossible for them to charge me retrospectively. I must have tried about 4 time to pay so personally, I have a clear conscience! It's just interesting how you paid with Amex as well.