Are you sure LAN told 200k miles? LAN uses kilometers, not miles.

Meanwhile, cheapest in theory and available for the date you need are two different things. Many programs raise the price when the cheapest is not available, so maybe there's simply no "cheapest" avialable currently for the dates you wanted?

LAN availability is best 11 months out for a couple of days, then most of their availability can get pulled.

And it's a high-demand destination, relative to capacity, so I think it's harder to get with miles all the way there than most other places in South America.

You may want to look into getting with miles to Santiago, and then paying money for a separate local flight to PUQ. (Just allow enough time to cover delays on the incoming flights, if you're booking separately, since you're not "protected" on separate bookings.)