Not sure if this is an option for you, but I successfully booked an award room for two. Applied a suite upgrade award, and then had Twitter adjust the reservation to reflect two adults and two kids, which they said was fine.

At the very least, it's worth a note to the Twitter team/call center to see if the base room can accommodate a child (specify whether the child will be sharing your bed or will need a crib or rollaway bed). If so, they'll be able to adjust the reservation for you. May as well lock in the award room for two right now, and then you can cancel if the hotel won't actually allow a child to share the room.