I prefer using BB plywood as it won’t split apart and is static enough to function perfectly for this.
The passage groove you cut in the back of the drawer box itself needs to be a bit oversized & rounded over too.
Sometimes people want a hidden stash section 😊 so doing a simple section in the back end with a separate and much deeper ‘U’-shaped cutout is an easy task. Note, you should dry fit it but finish it as separate loose pieces.
The surfaces can be wet sanded and final coated leaving all smooth surfaces, then a few small screws and you’re done.