Andrew Gabriel wrote:

I already have some ordinary plywood -- is there any varnish
which will make this waterproof enough to use outside without
peeling apart (as the current one has)?

Flowcoat it with epoxy (not polyester resin), then carnish it with
polyurethane. I've had a WBP plywood hatch treated that way on my boat
for the last twelve years, it still looks like new.