To head to S. America, I would think using AA miles to get to a US LAN hub and then use BA miles from there would get you the best value for the miles. BA allows a stopover in each direction, so by leaving from the US it should get you better stopover options. I don't know where you want to go in S. America, but I flew MIA-LIM (stop)-ipc (stop)-scl-eze(stop)-lim-mia for 80k in biz and $150 in taxes and fees.