Still no joy from Qatar. They responded with a suggested itinerary, complete with flights and dates, (they had changed from my specified route by replacing a third HKG transit with CMB). I agreed and asked for fare and payment instructions. They said (several times now) they're working on getting it priced and booked. They say the latest road block is the DFW-ANC-DFW flight number which is the same both ways. This makes it seem to violate the rule that you can't fly the same city pair the same way more than once. Haven't heard anything now for two days.

As suggested by pandaperth, I changed my booking tool entry slightly with first flight CPT-JNB:
(CPT-JNB-DOH-BUD-DOH-LAX-DFW-ANC-DFW-JFK-HKG-KTM-KUL-ICN-HKG-DOH-CPT)
After getting the "itinerary complete, proceed to book flights" message, and booking all flights in the booking tool, I was met with an error message that I needed to pick different flight or fare. They did not specify the problem. So, again as suggested, I tried it in L and A. Got same message. I tried deleting the DFW-ANC-DFW to see if they had the same issue as Qatar, but still didn't work. I tried several permutations of the Asia flights to no avail. However, when I replaced the CPT-JNB-DOH with CPT-LHR, I did get an ok and instructions to call BA in JNB to book. IIRC, they priced it at 78KZAR about 24 hours ago.
Any thoughts as to what the problem might be with the initial itinerary?
My plan is to contact Qatar once more to see what they say next. If no luck, guess I'll go with the flights through London.