Wirelessly posted (BlackBerry: BlackBerry8530/5.0.0.1030 Profile/MIDP-2.1 Configuration/CLDC-1.1 VendorID/417)
Neither AA nor BA permits free stopovers on such an itinerary. Best to use AA miles for each leg; just avoid flights on BA metal, because of the high surcharges.
The aa.com website cannot display award-seat availability on CX (or JL for HKG-TYO-ORD), so just call AA and ask.