I suspect that your greatest problem will be with ticketing rather than baggage. If AF and FJ have no interlining agreement re luggage, they are unlikely to have a ticketing interlining agreement, so I doubt that your ticket could be issued on AF paper. That said, it might well be ticketable on AA paper.

When you check-in at LAX, see whether they can check it all the way through. If so, then fine. If not, see how far they can check it through and see whether it would be more convenient for you to collect it and re-check in CDG or NRT.