Have your tried the flexible dates feature on AA.com, plugging in Destinations in S. America?
EZE, SCL, etc ?
For example, I found:
28-DEC: DFW-x/JFK-GRU (depart 12h40, arrive next day 10h40)
30-DEC: GRU-x/JFK-DWF (depart 11h35, arrive next day 10h55)
12,272 miles. One day in Brazil. One EXP renewal. 1255.20 USD.
I didn't have the time to poke around more than this, (to try to find something with a shorter stay), but maybe already this is worth it...
Good Luck.