Do you see any problem putting decking type screws between slats when making a cutting board. This would be done to help hold it together while glueing and for added strength and resistance to bowing. I seem to have trouble when I just clamp everything. It gets bowed.

Bad idea. If the stock sides are parallel, the parts could be creeping and sliding. You could try sprinkling a tiny bit of sand between the boards to prevent slipping. Also, as has been said, use cauls to hold the top and bottom flat.