So: arrived at SAA check-in this morning with three bags, each just under 32kg. The third bag was flagged as chargeable by the check-in agent's computer, and indeed, I paid just ZAR 300 to get it all the way to Sao Paulo. It's a nuisance charge, but not an unreasonably onerous one.
One weird thing though: the ticketing desk agent that took my money first looked in an old, dilapidated 3-ring binder, where she found a photocopied sheet listing United Airlines baggage allowances. For reasons unclear, she was convinced that since UA issued my ticket, their baggage allowances would govern my journey, rather than the published allowances of SAA, the operating carrier. And so, on this basis, she told me I'd only be allowed 2 pieces at 23kg each. Needless to say, this was wildly incorrect. Luckily, my fast thinking check-in agent simply implied that the only issue was my extra piece, so we didn't need to discover how much this erroneous idea would have cost me.
I still don't get how/why both Business Class and Economy have the same allowance...