IMHO your routing is valid

The relevant rule is:


I suggest you talk it through with them, perhaps as follows:

• Your travel is originating in Africa
• Therefore one of your visits to Europe/Middle East is required to be a transfer without stopover, which your second visit is (LAX-xLHR-JNB)
• The second part of the rule is not relevant to your itinerary, because one of your visits is from Africa to the Middle East (MPM-DOH) and the other is from Europe to Africa (LHR-JNB); for the restriction to apply both visits must be to/from Europe. Therefore you are allowed to include South Africa.