You might consider a DONE3 (Oneworld Explorer) with a route like LAX-ORD-MAN-LHR-BAH-LHR-XXX-HKG-XXX-XXX-XXX-HKG-JFK-DFW-SAN-XXX-XXX-LAX, then buy a separate RT on Gulf between BAH and RUH. The nonstop BA flight to BAH would require an overnight at BAH; via DOH you'd get in after midnight. Then the morning GF flight to RUH, returning either in the afternoon or the evening. Then back to LHR and onward to HKG - lots of miles and points. "XXX" means optional segments within the rules of the OWE, e.g. MAN-LHR-BAH-LHR-FCO-HKG...

You'd also have 5 remaining segments in N. America (two stopovers and one transcon) and can't pass through LAX or co-terminals (hence SAN.)