My cat vs. my carpet. How can I prevent this damage?

Oh no, not another cat question.
I hear ya, but this is more of a carpet question.

My 14 year old cat is still quite active and his favorite activity is to lie on his back on a step and tear the crap out of the edge of the step above him. He’s shredded three stairs worth so far. He only does it when I’m not in the room and doesn’t do any other damage in the house at all, so squirting with water doesn’t work and declawing is not even a consideration at his age.

My question is, if I can repair this without replacing the entire carpet? Would it work to cut a patch from an inconspicuous spot like a closet floor and sew it in to the damaged area?

And, if that works, do you have any ideas to keep him from doing it again? Are there any stair-nose protectors that aren’t a tripping/slipping hazard?

Don’t suggest a scratching post. He has one. It’s pristine.