Originally Posted by skimthetrees

Did you try my other suggestion? Call in again and ask to check availability from BKK-MEL without mentioning anything about connecting in SIN and without feeding them any flight numbers and hopefully this will search married segments and pull up the flights you want.

ETA:
The only other thing that comes to mind that might be affecting results is if AA has a problem searching married segments in this case because their programming mishandles F class married segments.

Hmm, another idea came to me. Find another married itinerary on CX that has availability with the first segment the one you want and book it. E.g., book BKK-SIN-HKG on CX in J (other routes to try, BKK-SIN-KUL, BKK-SIN-CMB, possibly BKK-(CX)-SIN-(AA)-U.S.A.). Within 24 hours call to change for free and replace the SIN-HKG with SIN-MEL in F. That way they cannot tell you there is no space on BKK-SIN because you already have it. They can search SIN-MEL and see it is available. If they cannot marry them together they have a problem they need to figure out.

ETA 2:
Perhaps try the AA Australian reservation number?