Your desired destinations maybe on a line - but it is quite a squiggly line. It also runs east to west with many points being roughly equidistant from Miami, rather then in a single line (or even two lines) going to and from Miami.
So the standard way is to do an open jaw, or return with different paths. But your problem is that all your destinations are spokes on the LAN network, with only a connection to one hub, although IPC has seasonal service to Lima as well. Since no backtraking is generally allowed, any one of those would be a final destination, and you wouldn't easily be able to get more than one in. You may have to spend some cash to do side trips (eg domestic GIG from IGU rather than IGR, or get a south american pass to do several flights), or do something like three single trips - but even then I don't think you'll all of these in.
You can use the oneworld map to help you plan - click on the three little lines next to the airline select drop down, select the 4 LAN airlines and keep the codeshare button ticked.