Good LAN fares to destinations in Peru and Chile that route JFK-GYE-LIM-xxx on the outbound and xxx-LIM-JFK on the inbound. It's business class (lie-flat) as far as LIM, and then the intra-South America flights to xxx are in economy -- though depending on the dates and destinations some of the legs can also be premium economy or business.
I've found a few options by mucking around in Google Flights, but I can't seem to construct an effective Matrix search. Maybe you'll have better luck.
$1,196 to SCL
$1,304 to COR
$1,321 to CUZ
$1,322 to LSC
$1,341 to ROS
$1,551 to PUQ
If credited to Alaska, LAN business flights earn 125% EQM and 225% RDM. LAN economy flights earn 100% in most cases.