Well, only CX and QF out of the oneworld airlines fly to Indonesia. DPS is Bali's airport. Best routing is NYC/HKG/DPS/LAX/NYC (or SFO) all CX apart from the LAX-NYC, with a 2-3 hour connection in HK. If you want CX metal on the return leg back to NYC, need to overnight in HKG.

Don't know about cheapest though. The alternative is going the other way round to SIN or HKG via Europe, then a CX flight to DPS from HK, or buy a SIN-DPS ticket.

Can recommend the Meridian Nirwana Bali. Plus the nasi goreng at Made's.