I'm sorry for your troubles. I would have thought that BA would be "in control" now since it's less than 24h from departure. What did the BA agent say when you called?

On the other side, I don't see AA easily dropping the CPH-LHR flight unless you have the luck of finding a very sympathetic and open minded agent.
Have you researched any alternative plan beside asking to drop the first leg? Are there other flights available to you to reach CPH before the next operating OneWorld flight out of there?
Could BER and HAM be better alternatives? (they are within 300 miles). Or could you move the departure by one day or two?