You are right. BA is wrong.

There is no restriction on the number of stopovers at any one point in an itinerary. The rule you quoted is not relevant - it does not refer to stopovers; you arrive in the continent from Africa and depart to North America, this has nothing to do with stopovers IN the continent.

At the end of your itinerary how are you getting back to Africa? If that is through Europe/Middle East then you might have a problem depending on the itinerary - your return must be a "transit without stopover" and must be through the Middle East Zone, not the Europe Zone (because you have included South Africa in the itinerary).