I've made my itinerary, more or less, with the help of a couple of amazing reservations agents at Star Alliance.
I couldn't get myself out of Tahiti anytime between early December and Christmas. So I am going to try to get to Easter Island from South America instead. The run from Tahiti to Lima is pretty brutal, and then I'll still have to get to Santiago after that.
Click here to see what is almost certainly what I'll end up doing. Unless I can get a ticket later out of Tahiti down the road, like if more seats are released.