What could one spray inside solid core door to protect wood and create a tunnel?
I think if you use a couple coats of good primer and a finish coat of enamel paint, you should be fine. Just be sure to work it into the wood well. Two thin coats are always better than one thick coat.  The cat door is gonna protect the cut edges quite a bit as well. Put a small bead of silicone caulk on the inside of the cat door bezel before you install it on the outside surface to seal that possible entry point from rain water.