Yes. I have done this many times. When you check in at the AA counter, just tell them you are connecting on to a CX flight and give your final destination. They will also through check your luggage all the way on to your final CX destination. you will NOT need to collect it in LA. Generally they will NOT issue you your CX boarding card though. you will need to still stop at the CX desk in TBIT to pick up your CX boarding pass. The Admirals club should be no problem to let you in. just show them your eticket receipt for the CX flight. Although the Admirals club does close (I think at 11:30?)

The only caveat to the above is if you are travelling on a US passport with no HKID on a one way ticket to HK. Technically US citizens need a return ticket for travel to HK. AA is a stickler for this and will not through check you to the CX flight as they could be held liable for your return costs. CX could care less (as well as HK immigration) if you are on a oneway ticket. Even if this is the case, just tell the lounge you are connecting to CX, show your onward ticket and it should be no problem. They are generally good on this.