even if you want to goto NRT, I think you can still do DPS-HKG-NRT-HKG-LAX. You have 4 segments in Asia.
About EXP, is the "35000" you currently have Q-points or Q-miles? If they are Q-points, you can easily get EXP with your OWE, otherwise if you goes for miles system, you may need to do some turn-arounds to get more miles, such as LHR-DXB-LHR-IST-LHR, etc.