IF CX is doing the reissue and they say the flight does not exist, you have to go by their rules. If BA says that the flight exists, then you have to go with BA's ticketing. You can't convince CX that the rules exists in BA's system as they can only go about what they have in THEIR system.

AA's office is in the same area as CX's - just next door. If you can get AA to interpret the same rules as BA's, then get AA to reissue the ticket even if it is a paper ticket.