Hi allset2travel - good luck with your rtw - have done 5 of them ex CPT in the last few years and they can be a bit complex to get the hang of.

Firstly, what you are looking at with your itinerary is a DONE6 - as your departure continent counts as 1 and even though you are only transiting Europe it still counts as a continent.

As Pandaperth points out you can transit Europe twice ONLY if you are ending in certain Africn countries - like Kenya, but I assume that is not where you want to end up. You can however fly from HKG to JNB on CX to end your journey. however to do his you would need to reduce one of your Asian sectors to keep within your 4 flights allowed in one continent

Looking at your routing I gather you are trying to maximize your miles as well. The following is a legal routing - but still a DONE6 I'm afraid!
JNB-xLHR-xLAX-SCL-LIM-GRU/EZE-xJFK-SYD-PER-HKG-DEL-NRT-SIN-xHKG-JNB

This effectively gives you 15 sectors so you could add another flight somewhere. As you are paying for the continents anyhow you can make stopovers in Europe and North America (only once the second must be a connection)

If you use LHR as a pure connection using BA metallic on AA code share and the onwards connection to the US on AA metal you can ticket as AA. I think the reason that it would not validate the xLHR was because you tried to transit Europe twice which is not valid to start and end in SA.

The other option to avoid the ridiculous BA and Heathrow taxes and fuel surcharges would be to use IB to MAD from JNB. IB are actually not that bad in business.

Hope that helps!