We just recently purchased USH-AEP tickets for the same reason, no availability on LAN flights.

I did notice the price differences, mainly Argentina vs. rest of the world, and booked the ticket from the german site as it was priced in euros (my currency as well, Finland). The confirmation for the reservation stated that "The management fee included in the quotation applies only if the ticket is issued in the country of the website you are browsing."

The online reservation did not offer any options to pay for the ticket and I had to send some e-mail around. Aerolineas office in Germany replied that I could visit their office or call the call center in the US to pay with American Express. Aerolineas from Argentina forwarded my question to their office in Sweden, I sent them my credit card details and it was charged in Swedish kronor. There was a small fee added to the German euro price.

I guess you could do the booking e.g. on the US site and call their US call center from Thailand for the payment?