If you fly to SIN on UA, you can also look at a LCC like JetStar or AirAsia for the segment to DPS. I suspect they will be a lot cheaper than SQ. Both depart from Changi Terminal 1. I've taken both and they are both just fine for a short flight.

You would have to collect bags and check in separately.

I do this all the time (to other locations). Max out UA metal and miles (and SWU!) to SIN arriving midnight, take a 6hr rest in the transit hotel, then take the short finishing flight the next morning, fully rejuvenated. Probably the cheapest option too.