The cat does own the joint. He will not change. You may be able to keep him away from the door when you are armed and dangerous and on the same side as he is, but when you are on the other side of the door, forget it. He isn't scratching there because it's carpet like his scratching post, he's scratching because he wants to get through that door, dammit!
Put mats on the carpet, against the door, on the cat side of any door that you plan to close. If the cat scratches the dickens out of a mat, you can toss it and get another one. It's much cheaper than replacing a carpet. Vigorous scratching may move the mats away from the door and expose the carpet. Pin the mats down if you have to.