According the aa.com page about non-traveling security access, a non-traveling non-member can get access through security if they're attending a reserved conference room meeting at an AC. If the family member happens to be an AC member, then they can get through without booking a room, just by pre-registering.

So basically, you need to book a conference room at the RDU AC, then pre-register your family member who will be bringing you the Bojangles.

Here's another similar thread. If you could get in touch with the RDU AC directly (I miss the good ole days), I'm sure the AAngels wouldn't have any problem registering your family member to get through without the trouble of booking a room...especially for the business critical Bojangles delivery.