1. Who booked the res for you? If your agent, then he/she should be able to do seat selection for you.
2. The flight in particular , CX7565, is actually AA607. CX cannot display the seat map on their system, whereas AA can. You can call AA to find out what seats are available, then call CX to request that particular seat on CX7565 since CX cannot display the seat map. But they may request for a particular seat without displaying the seat map.
3. If all else fails, do a generic seat request - ie window or aisle through CX.
By the way, if you're hoping to use AA 'stickers' to upgrade the ORD-LAX flight to F class, well you can't as it's a CX flight and not on AA document anymore.
I really don't see why you couldn't have booked ORD-LAX on AA, it would have been so much easier overall.
As an AA Plat anyway, once ticketing is done, you may also get seat assignment on CX flights.