If you open your reservation, it should have both an AA and a JL confirmation number (this may be different than the record locator). If you can locate the JL confirmation number (usually listed below the flight number for that leg of the trip), you can use it on the JAL website to select your seat. This worked for me on JAL in January.
See below example for an upcoming trip I booked through AA codeshare QR - it looks and works the same when codeshare with JL: