christep, thanks for the info re: rerouting.
If I change the itinerary to the below, will I be able to book it? It looks to me that it should work, with being charged twice for having too many sectors/segments (one too many sectors in South America, and one more segment than 16), but I may be mistaken. The online tool isn't helpful for this because it just gives me the too many sectors and too many segments error messages.
JFK-EZE-UIO-SCL-IPC-xSCL-LIM-xLAX-NRT-GUM-TPE-HKG//LVI-JNB-CPT-LHR-NYC