Tough situation.
If there are no options for EZE GIG then
Wait for Economy to open for EZE-GRU on JJ or Qatar.
If you are flying Business, there are seats on 12/11 already for both airlines.
Since both JJ and Qatar participate in AA you can call agent who should be pretty nice about getting you on EZE-GRU-GIG or perhaps another routing, but there would have to be existing space as they cannot force award space to open on partners.
Technically LAN should just rebook and find a way to make this work, but that's a lost fight. That's all I can think of.
Or you can find another routing EZE-SCL-GIG?