I'm having a similar issue. I (American) tried booking a flight today for my mother-in-law and her daughters (Jamaicans) to Panama and then onward to Cancun. After booking I received an email saying that the tickets will be processed within 6 hours.... of course it wasn't... I called Customer Service and actually got through very quickly. The agent explained to me that the purchase was rejected because it was flagged for possible fraud since an American card was used for Jamaican passengers.
In this case... the flagging does make sense because there is a huge rise in scamming, particularly in Jamaica, so I can understand why it would be flagged. Similar to what you all have been saying, the agent told me I just need to submit ID of the cardholder and the passengers, and then provide some sort of bank statement. Normally this would scream scam, but I can understand why it would be necessary in this situation.... thoughts? Will it be safe to submit a bank statement?