Dogs must remain on the car deck, either in your vehicle or in a carrier until the ship docks.
http://www.dot.state.ak.us/amhs/pets.shtmlYou can't stay with your dog and your dog can't stay with you. But visits are scheduled for humans to go to the car deck to feed and help their dog with its business. Usually no more than half an hour every few hours. Most dogs do their business on the car deck. You can also take your dog out at ports.
On Edit: the link above indicates, "...there is no charge on any route for a certified service animal traveling with a person with a disability.
Service animals do require prior approval from the Traffic Manager so that shipboard staff can be properly notified."
Wonder if your dog might enjoy a flight from Anchorage to Seattle more? Then it is just a few hours of time and your dog will be done with travel. Perhaps an Alaska friend can put your dog on a flight to coordinate with your arrival in Bellingham?
Propane must also be turned off so plan your fridge accordingly.
