They are wrong. Call the AARTW desk again and tell they to get it sorted out or you'll book it from someone else. The rules are very clear. There is nothing in them about the country of issue or the billing address of the credit card.

What exactly do they mean by "country of origin" of the credit card holder anyway? Are they judging this by the issuing bank, or by the billing address (which could easily be in a different country from the issuing bank) or by your passport or what?