Now this solution is pretty smarmy and I haven't tried pricing it out to see if it will actually work in practice, but the theory is GREAT.

On your third segment, do ATL-LHR and book the last segment on BD for a future date when you can use it on local business. Then just buy a separate BD ticket for your flight home, in fact make it a roundtrip to get you back to LHR - those fares are extremely cheap if you can get one of the GBP40 specials.