Potential dates? I would go LHR-MCT (or DXB/AUH etc., really anywhere in the Middle East is way more miles than DME but MCT is the maximum)-LHR-MAD-AMM. Since you didn't go AMM-LHR to begin, you can fly anywhere in the middle east and then back to LHR, then just go to MAD so you can get back to AMM. May be able to pick a place other than MAD, though I think MAD is the max miles you can get LHR-XXX-AMM in Europe. If you don't want to go to the ME, then DME is probably your best bet, but LHR-MCT-LHR-MAD-AMM is a few thousand extra miles over DME.