Maybe somebody else with more knowledge of the internals can chime in. But from my brief research, it looks like APIPA relies on the use of ARP to figure out what addresses are already in use, but it appears that there is no way to access the ARP table of the CC3000 (it is only used internally, and not exposed).
I’m not sure if the CC3000 will fall back to APIPA on its own if DHCP configuration fails. Have you tried it? If not, you could just try assigning an IP, but I don’t know how you could detect/avoid collisions.
Anyone know more about this subject? @Dave?