You can buy baltic birch 1/4" at menards they have 2 different choices 1 is floor underlayment its 5 ply the only problem with it is that it has nail markings so it would nt work well if you were going the wood look. The other is 1/4" also baltic birch 5 ply and seems no different than the 60x60 inch baltic birch the specialized wood lumber carriers have. Ive tried the plain baltic birch from menards making a 60" dancer longboard and know it works and is quality (wouldve used 60x60 piece from local lumber supplier but needed extra inch on both ends for trimming). The price for the 4x8sheet was $27. MAKE SURE ITS BALTIC BIRCH!!!!!!!!!!!! So your buttom layer on your plywood core is possible to make in 1 piece.
Only problem is the 1/4" part. I'd rather be working with the 3 ply, 1/8" baltic birch I get from Exotic Woods. The stuff is great, no voids AT ALL, just beautiful woodgrain. If only it came in 4X8 sheets...
The 1/4 " is void free and made the same way the 1/8 " it would also be better to use 1/4 for the fact that you would less plys to lam together for fear of delamination.