Just think about the magic stringer for awhile… even if you don’t come up with the same solution we did, you’ll come up with one that works. The concept is to control the types, locations, and amounts of flex - independently on the deck and on the bottom.
I also found that making the deck stiffer than the bottom worked great. We made boards where the deck mostly controlled the boards overall flex. The bottoms were much more flexible - we called it “dynamic bottom technology” or something like that. You could actually see the bottom change shape in turns. The rails were stiffer so you could bury your edge and hold through turns, but the bottom of the board would adjust to the water pressure under it. I think this really helped with speed. The board could match the shape of the water beneath it to help reduce drag. The concept looked and felt like it worked. Guys said they had their best waves ever on those boards. They worked for me.
As far as your stringer goes… I think balsa would surf better than foam. Just make sure it’s not too brittle. It will get a lot of stress. When we used stringers, we made them from fiberglass - shaped I-beams. I forgot exactly what glue we used but it was made by 3M. It was flexible, super strong, resistant to salt water and UV, and really expensive. We used the same glue for joining the halves too. I think we spent about $25 on glue per board (at volume rates). The adhesive comes in a double tube - like a 2:1 ratio. You need a special gun and mixing tips to dispense it. Hand operated guns are around $30. We had pneumatic ones that were around $600 each. If you’re just gluing the stringer (and not the halves) I bet silicone would work. You’d just have to make sure air gets circulated inside the board so it will cure.
I’d really like to try some boards made with end-grain balsa. I bet they would work insane.