Travel dates are right around Lunar new year so it will be tough to find deals.

You can redeem your AS and AA miles for CX travel so that's where I will start... LAX-HKG.

Once in HKG, you can find rather cheap airfares with LCC or even legacy carriers to TPE and PVG (although Lunar new year thing may pose some challenges).

Suggested itinerary: LAX-HKG-PVG [high speed train] PEK-TPE-(HKG)-LAX

The return leg thru HKG is only if you are redeeming on CX. Otherwise you can find TPE-LAX fare separately from CI or BR. Don't forget CI also flies TPE-ONT so that's another option.