Depending on how much time/miles you want to spend on the journey, another routing from LAX - DME would be on CX via HKG. You could do it much faster via LHR, but if you're allergic to BA's high fuel fines and haven't enjoyed CX's fine TPAC service, it might be worth investigating. (The HKG - DME only runs 3x a week, with A343 equipment with a good business class cabin up front.)