You're best bet is probably AA since you have the most miles there.
I am not sure if you can do this on AA miles, but I know this can be done on UA & DL miles.
1 round trip ticket with open jaw stopover:
NYC - SIN - TYO (open jaw stopover) ICN - NYC
1 one-way ticket:
TYO (NRT/HND) - SOL (GMP/ICN)
If I could, I would fly:
CX F: NYC - SIN (via YVR, HKG)
JL J: SIN - TYO Direct (Last I checked no F on those flights)
JL F: ICN - NYC (via TYO)
since TYO to SOL is a short 2-3 hour hop, just rough it in Y.
all the best.