In the US, zip codes are often used as the verification method for credit cards (i.e. if you were using a debit card it'd be a pin number.) So getting the zip code correct can make or break the transaction.

Have you tried entering your current billing information for the card and adding zeros to get to a five digit number for a zip code? Or is the card being rejected? If so I'd try and use a US address if possible, the only down side is that any mail you receive from AMEX will go to that new address. Otherwise you shouldn't have a problem.