Sorry Dana, Looking back at my first answer, I simply said "No" and you might have overlooked that reply. If you think about it would you imagine that a Belizean visitng the US would get his tab charged in Bz$?
Now maybe there is some special international card merchant account operating in USD that has such a provision but the merchant is probably going to have to pay (dearly) for that customer convenience and they're already getting nicked a xaction fee of 3 to 4% right off the top for the "convenience" of accepting your card in payment. The chances are slim that any merchant in Bz, (lol, other than like, maybe, Francis Ford Coppola's places), are going to be doing that.
Based on our experiences and the explanantions given to me, the merchant's credit card account/bank requires that the charge be made in the currency of merchant account origin. The banks are (now) required to apply the prevailing daily rate of exchange to in any currency conversion (thanks for reminding me of that kreh) so they have to see original currency charge. Now you might see a bill presented for your convenience in USD (based on the merchants interpretation of 2:1) but the charge is made in the unadulterated Bz amount and forwarded from the merchants bank to the cardholders bank for "legal" conversion.
If there is one constancy it seems to be how banks can work together to make money off the card holders and merchants on both sides of the "fence" and either of those parties will pay extra for any additional "convenience". Maybe call your credit card company and ask the question for a definitive answer, but I already suspect what the answer will be.