You're being given the runaround.
The terms of the ticket (contract) as bought are that the ticket is valid on any date/time on the route purchased (subject to using the coupons in order and so on). The technical limitations of the airlines' systems (real or imagined) are not your problem. You have a contract which the airline needs to honour.
The ticket does not need reissuing. I would simply get the reservation changed and then call American to make sure that they can see the ticket number in the changed reservation. You should get the AA PNR from CX or your TA to facilitate this.
CX's rules as published to travel agents may be downloaded
here (.xls format). They state explicitly:
Rebooking
Before departure
Permitted without charge, the fares and charges shall be calculated
in accordance with the fares and conditions in effect on the date
on which the ticket is issued.
After departure
Permitted without charge