I don't have any specific advice as I've never sent a pet as cargo/checked luggage, but I would recommend that you do not let up on the calls/information seeking until you have a satisfactory outcome, be it a good enough update on the process or when the dog is in your possession. I would recommend always staying calm and professional, even though I can imagine this may be a trying time for you.

With any luck, perhaps NW scoop can look into the matter. Maybe not in real-time, but perhaps follow up afterward and figure out what went wrong and how they can improve their coordination with their vendor Priority Pets.

Best of luck.