This may be to late for JTG but hopefully not, and hopefully a reminder...

If this ordeal is your re-entry into frequent flying, then at least you should try to get something lasting out of it besides memories of BA cattle class.

American code-shares on the BA LHR-JNB-LHR legs, if you fly AA transatlantic. While BA won't let you join the BAEC with a discount Y itinerary, AA will be happy to let you challenge on AA-numbered flights. If you're not familiar with AA's "Platinum Challenge" then go to the AA board and read the stickys right now.

Looking on ITA, an itinerary booked through aa.com going LAX (AA) JFK (AA) LHR (BA) JNB (BA) LHR (AA) JFK (AA) LAX for your dates is almost exactly the same price as the BA flights. You'd be adding a connection at JFK both ways, but honestly in coach I think that's a good thing; time to stretch legs and walk around a bit. Plus, AA's 777s over the pond are set up 2-5-2 vs. BA's 3-4-3 in Y, so one less middle seat per row and the same number of aisle/windows.

Here's a link to an ITA fare sheet showing a possible itinerary. http://tinyurl.com/nj7kn By Plat challenging, you'd hit Plat on the LHR-JNB flight, and from that point forward you'd earn double miles. I estimate for the whole trip you'd book something like 40K miles counting Plat bonus. Plus, when you get to LHR on the way back, you could most likely explain your situation to the Admirials Club people and get lounge access at both LHR and JFK on the return legs.

This may delay your arrival at JNB or back at LAX by a few hours, but IMO it's seriously worth that in terms of ongoing status, greater comfort in flying, and making the most of your company's (and your own) investment.