I've got shredded carpet in several doorways from the same thing, LOL. When they want in or out of a closed door, they really want in or out of a closed door! My guys accidentally shut themselves into my home office once and my brand new carpet paid the price. Havoc was probably playing with a toy between the door and the wall and pushed the door shut from inside.
Cats don't like the smell of mothballs. You might want to put one near the door, if you can stand the smell yourself. But it might be too strong for kitty's nose to be in the house at all. Then again if you've already tried other repellants, this might not work anyway.
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