I honestly don’t see how your passport would matter, particularly when under Argentinian law a person born in Argentina is Argentinian irrespective of which passport they use to enter the country. I don’t know if this is still the case, but years ago immigration officers would allow you to enter the country on a second passport but the moment they saw the place of birth was within Argentina, they would write “ARGENTINO” on top of the entry stamp.

If you use a foreign card VAT will be waived, at least in my experience. In the odd case when somebody would ask, VAT is based on residency and you can always explain that you are not an Argentinian tax resident.