You could move UR and MR to BA, but that's not a good choice unless you are booking a single segment trip. BA charges by the length of each individual segment. I used BA miles for my outbound, LAX-EZE. Because it is one sector, it was 30K for coach or 60K for business, plus $5.60 security fee. The return will have something like an $80 fee due to Argentinian fees. I believe it would cost 50K for one way on AA in biz and 45K for one way on AS. Those carriers allow multi segments. You will need to call to book should you choose AA. Just last week, AA waived telephone booking fees for any partner that cannot be booked online. If you see space that suits you, I urge you to grab it. You want to fly back from South America at peak time. Award seats are rare and valuable. I know when my return LAN flight should be available, even to the time of night. I will literally set my alarm that night so I can get online and get the seats I need. I have never done that before, but I have no flexibility on date of travel.
When you are checking availability, also look at LAN from SCL-SFO. You might find award space from GIG-SCL, then SCL-SFO.