I tracked this route for several weeks earlier this year. I discovered that LAN releases very few redemption seats to other OW partners, even those they show at the lowest award levels.

I found occasional seats on ba.com but nothing close to the dates I wanted. If you search here you'll find threads from others similarly frustrated. I ended up booking cash tickets - booking on the LAN Chile site was much cheaper than LAN UK or US. We got a really good deal - r/t in J was about £500 as I recall.