Wirelessly posted (BlackBerry: BlackBerry8530/5.0.0.601 Profile/MIDP-2.1 Configuration/CLDC-1.1 VendorID/417)
The only awards seats on AA or AS that are bookable with Avios are "SAAver" seats that show up on aa.com. AA makes AAnytime awards available to AAdvantage members only (for twice the price of SAAver awards), but does not make AAnytime awards available to BAEC, or to any other frequent-flyer program.
The OP can try splitting his party up into two or three groups, in hopes of finding a couple of SAAver seats on one flight, a couple more on another, etc. But June is a peak time for Hawaii travel, so finding SAAver seats might prove challenging.