All of the previous points are great. The other thing to consider is that not all birch plywood is created equal. I have had similar problems with imported birch plywood. You may pay a bit more for domestic but the veneer is usually better quality and it won't peel as easily. I've noticed the imported ply has gaps in the plys after cutting. A mdf hybrid core is an option because the ply will resist warping. Good job on your project.
Hi....another direction for your consideration as most of the solutions have already been given. Since all my projects are constructed of birch ply being able to get 2 very good sides and more stable (additional plies). If you can, try to acquire real Russian Baltic Birch plywood 5 x 5 as it has a much thicker finishing layer than standard 4 x 8. We call this stuff Quebec Baltic Birch and avoid it unless being painted. The thicker venier stays in place and you can actually sand it