Piper's idea sounds like your best bet. Trying to straighten the backing by gluing it to the core is asking for a warped or twisted bow.
I have noticed over the years that wood has to acclimatize to the area they are in for a while before sawing. Even then you will get some movement. This it where the difference between dry wood and cured wood is so evident. With well cured wood, movement like this isn't as likely.