I'm not sure about availability, but if you're already getting the Brazilian visa, you could consider this:
MIA-LIM-IGU/IGR-AEP on LAN (stopping at each destination), then buy an AEP or EZE ticket to GIG, then use BA miles for AA GIG-MIA. This avoids the Argentina fee because your entry point is over land at Iguazu Falls, which is a very nice destination in itself. Another alternative would be to bag Buenos Aires and do MIA-LIM-IGU on LAN, then buy IGU to GIG, and miles for GIG-MIA as before.
I don't know that this final alternative would work, but it would be a true coup if it did:
MIA-LIM-IGU/IGR-AEP-MDZ-SCL-GIG all on LAN, with a stopover everywhere except SCL. Avoid all fees (except Brazil visa), but I think it severely violates "direct routing", and is predicated (as is the first itinerary) on IGU/IGR being considered co-terminals.