How willing are you to make this a bit more roundabout in order to save a ton of bucks?

Latam (Oneworld, former LAN Chile + others) is selling Miami - JNB via Sao Paulo (GRU) in business class for around $2000 less per ticket ($2717) than LH/EK out of DFW. You could use your miles to position to MIA, and stop over in Sao Paulo on the way to or from (might cost $100 for the stopover; the rules are a bit vague.) If you stopped over at GRU you'd need Brazilian visas. In flying miles it's 2000 - 3000 less each way compared to FRA or DXB, and because it's winter in the southern hemisphere the weather ought to be okay.