Just wanted to post that I finally got this to work (sort of). By searching on BA's site for each leg individually of all the possible flight plans for my trip from Chicago to Hawaii next year I was able to book 4 of the 6 possible legs using my Avios. Cost me about $80 in fees and many hours so far. I still need the LAX-ORD part, but figure this should be easier than the rest.
Thanks to everyone for their advice and patience.