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travelinterpreters, you will not get very good value from your BA miles on any of the three itineraries. BA charges more miles for redemptions that involve flying more than one BA partner.
If you could get you and your family from MCI to ORD somehow, you could use the BA miles to fly JL ORD to BKK (via Tokyo).
I would use the AA miles for the MCI-CNS and MCI-CUZ trips.
Don't forget about the free North American gateway stopover that AA permits on international awards, and the free stopovers that BA permits on award travel.