It's a shame what these airline budget cuts have done to the customer service they once provided.

BA may be reluctant to do your ticketing, as it doesn't appear that you have any long distance BA segments unless you are planning to use them over CX on the HKG-LHR segment. If they decline then I would go back to the AA counter at AMA and insist that they find a way to get you the ticket. The excuse of not knowing how to hand write a ticket is bull. A supervisor or the station manager surely knows how to do it or can find a way internally at AA to get it done. That is there job. Don't let them off the hook.