This might be a good way to beat them at their own game: Buy a roundtrip from the US to any close city in Central America, like Guatemala or Belize on American: 2 continents; Use a Cathay Pacific AWARD from the U.S. to Australia with FREE stopover in Hong Kong...THEN buy a cheap roundtrip Hong Kong to Taipai on CX for the ASIA continent, and buy a cheap roundtrip on QANTAS from say Sydney to another close city inside Australia, if domestic flights count, (not really sure!!), or buy a cheap roundtrip Sydney to New Zealand. For Europe and Africa, the Iberia Spain to Morrocco roundtrip makes sense...
ANOTHER SUGGESTION: Tel Aviv is technically in ASIA, but might be CONSIDERED "geographic Europe" by British Airways...
ALTERNATIVES: For ASIA, London to AMMAN on British, or Tel Aviv if it's considered "Asia"; Does QF still fly LAX to Tahiti roundtrip? What about Honolulu to Fiji? Also, what about US/Cairo on British, open jaw, and return from Tel Aviv or Amman to US, with stop in London...this MIGHT earn 4 continents...then just do a Qantas LA/TAHITI, AND a quick trip to Central America....Oh well, still seems like a lot of trouble!!!