Well, you could buy two separate tickets (ORD-NRT-ORD and NRT-SIN-NRT) and nest them at your own risk. Then, if you miss the connection, it's your problem, but AA stops telling you you "can't" make the connection. Obviously a risky option -- especially since it's not obvious what JAL will do for you if you do in fact miss the connection. But something to think about.

BTW, may I ask how much your fare from ORD to NRT is? Stopover permitted in NRT in one direction, by any chance?