I would fly LAN. Unfortunately, right now, the flight LAX/SCL is one stop, and the flight SCL/GIG is one stop, making for a slightly longer trip than desired. (It looks like the LIM/GRU flight that would connect to LAX/LIM is on a narrowbody and it is like euro-business (premium economy) so you'd want to go through SCL for the comfort the entire way.

SCL is a very easy connection, and GRU you stay on the plane. Any trip to South America where my first point of entry on the return isn't MIA is a good thing. You will end back home @ LAX after a nice 8 hour flight segment from LIM in a flat seat and then go right home.