While trickier/ tougher to mix cabins ‘in one direction’ (typically, you’ll pay the fare for the higher cabin), many round trip fares allow combinations on a half roundtrip basis - one class in one direction and another on the way back (significantly cheaper than 2 oneways in different classes)- eg biz in one direction and coach in the other..

Couple of ideas:
1) Try a search via ITA Matrix (something like /f bc=d for one direction..or maybe the other biz fare classes , and ‘cheapest’ as the overall search parameter). Call in to book if any of the power tools links doesn’t work.. (needs an understanding/read through of ITA matrix)

2) Also worth a try, search for the lower cabin on the website, and in one of the directions, choose a higher cabin if offered.

While I never ticketed, a while back, I did find US-SIN-XXX//XXX(or, YYY)-SIN-US fares with the outbound in J, return in a lower cabin..at prices significantly lower than US-SIN J fares (or US-SIN-XXX for that matter) using both the above….