I think I found my own answer, more or less. Per the BA rules for award bookings:
13.8 There are no charges for stopovers included in an Award itinerary. Where stopovers are at connecting points on the most direct route between origin and destination.
So if I book the trip as a pair of one-way awards (ORD-EZE, then EZE-MIA), I can take a stopover at either Santiago or Lima on the return, since those are connecting flight options BA.com presents. Obviously a phone job, but with flight numbers in hand and availability showing on BA.com, it shouldn't be hard.