A registered friction fit on 1/8 Baltic birch is very rigid and would likely be moreso if you glued it too. You could try cutting 1/4, but 1/8 cuts much more reliably. It’d be interesting to rigorously strength test a 1/8 assembly versus 1/4, but who has time?
Also, apparent rigidity is a function of size of your piece. 1/2” square of acrylic feels like a rock. A full sheet bends like a banana.