Surprisingly, BB is not available here in town. I imagine Paxton could order it. Someone else recommended MDF core. I'd have to use thread inserts but it would durned sure be smooth and flat. The veneer I see available is 1/32 and the US-made plywood claims to be 1/32. Am I missing something?
If you are going to lay up your own veneers, baltic birch would be a far superior substrate to the maple ply. The maple ply only has a maple face, the inner plies are likely poplar or another cheap/soft species. baltic birch has all birch plies.