I know that there is a water activated gorilla glue you can use to bond styrofoam to wood (you actually have to wet the surfaces to activate it). I don't really know how good the bond is. And the glue ain't cheap. I have used it before and it is suitable for my purpose but if I were you, I would test the bond before I went ahead with using gorilla glue. There may be others, I don't know.
This brings me to a interesting thing I will point out. Some people disparage the foam/wood laminate type construction. I will tell you first hand that you can make some amazingly strong structures with this type construction that are also amazingly light weight. There is nothing wrong with this construction method as long as the bond holds.
