Remember to read the small print on any sale fares offered on the airlines' pages in Argentina. Often they limit these fares to Argentine residents, and they require that you show documents to show it at the airport.

When I wanted to go to GIG I too used miles. A business class ticket is only 30k miles, making it a very good use of them. I took LH to GRU and then RG to GIG, and then RG non-stop back. If you don't want to use miles then I'd suggest trying to use Gol and playing with your dates. RG and TAM are not usually very cheap.