Defintely use some other search engine, e.g. ITA matrix, to find a route then force it as multi-segment wherever you are doing the actual purchasing. GRR is far enough east that you can also look at flying into LGA and transferring to JFK. Delta will provide a voucher for transportation to connect to/from an international flight at JFK and a domestic flight at LGA or EWR.

For a random time in March in ITA matrix I see GRR-LGA/JFK-NRT-HGK and back in Z booking class for $5244. Layovers in New York are 5:30/6:15 so plenty of time to make it through traffic,