OW Devotee:
Has your TA called BA to find out what their problem is?
I really dont think BA has the OWE rules straight.
Your itinerary does qualify as long as you stay less than 24hrs in LAX on your SCL-LAX run and continue to HKG. On your return from EBB you are going directly to NA/BOS and therefore you qualify there as well.
Tell your TA to send the rate quote over to CX. They usually know better.
Does your TA have Sabre? The entry for the up-to-date fare rules is N*/OW EXPLORER.
If you need more help email me directly...I do these RTW's constantly.