I've had a bad experience with JL. I had KUL-NRT-JFK and was putting the whole thing on AS. However, in Tokyo, I showed my BA gold card to enter the first class lounge instead of the business one.
When I looked a few days later, both flights had credited to BA including the one I had already flown before I ever showed my BA card. It used to be a pain to have BA remove incorrect credit but now they accept to do it on the phone, so I did it and then sent a request to AS.
All ended well.

In the OP's case, the AS flight cannot credit on AA anymore. So putting the AA# should not be too risky. AS will credit a posteriori if necessary.