Seems like your best bet would be to book the entire itinerary on United, transferring your UR points to your account giving you 210 UA miles. You could book the following:
Business Class - LAX ->ICN (Open Jaw), then BCN -> RDU
This itinerary would cost 110,000 UA miles and is bookable on the UA site using the multiple destinations function.
Then book 2 one-way tickets to get to BCN to complete the first ticket.
Business Class - ICN -> PVG
One way ticket booked online on UA would cost 20,000 UA miles
Business Class - PVG -> BCN
One way ticket booked online on UA would cost 65,000 UA miles
Seems to be plenty of availability if your dates are flexible and would cost 195,000 UA miles + taxes.