If you attempt a repair and decide to repaint the door, here's a trick that "might" help stop the dog from chewing the door again.
The short version of a long story. Years ago a good customer adopted a rescue dog that had an appetite for her kitchen walls below the chair rail. We fixed and repainted the area twice, after he chewed through the drywall. The third time my brother asked the customer for permission to mix Cayenne Pepper into the paint. The customer was game to try anything. Problem solved. It might help.