I somehow managed to pull it off on the Aerolineas website. First time I tried, when I put in my nationality and passport it changed the final price and added a tax. Tried it again the next day and it went through. Paid with a friend's Argentine credit card and got all my reservation details in my email. Probably would not be able to do this over the phone or in a local office.
After you fly can you do a follow up post? I know previously people have tried this and when they check-in at the airport if they can't present evidence of residency they've had to pay the difference before being allowed to board their flight. let us know what happens for you!