I would phone AA's ATW desk in the USA to arrange the flights and then pay through AA UK . The number for the ATW desk is +1 800 247 3247 . The UK office number to pay is +44 207 365 0777
A reason why the booking tool is not showing the flight could be that there is no availbility in L class on the day you wish to fly. Even so, by booking with the AA desk you are likely to save on fuel fines
Dave