I have no idea how you could find this, but there are these things that used to be popular back in the 70s. In the 70s stiff ski boots were the hot new thing. Many people at that time had old ski boots and couldn't spring for a brand new set. As a consequence you could buy a product called a, "jet stick," which was a plastic toe and ankle piece that you could strap on to the front of your boot and give you the feel of a nice stiff boot.
If you could find or make something like this you would have a light weight stiffener to strap on your climbing boot. It might give you more control for skiing yet be removable for climbing.