This should be doable:

LHR-LUN//JNB-HKG-SIN-HKG-PEK-HKG-SYD-SCL-EZE-GRU-LIM-MIA-MCO-LGW.

You'll need to get a separate LUN-JNB ticket, as well as GRU-GIG-GRU if you want to do Rio. You could do Tokyo as well by buying two additional Asia sectors on top of the four you get included in the DONE4, so you'd add a HKG-NRT-HKG trip into the mix. Until JL joins oneworld, you pretty much have to fly via HKG for everything in Asia. I'm a bit rusty on the details, but IIRC you have to fly via SWP to South America, so that means adding an extra continent. (Is or was a North American transit allowed for Asia-South America?)

You could further optimize your routing by adding five more segments in North America if you want to maxmize miles.