It is per character, so you’ll want to make sure that you’re on the character who found Dog in Pandaria, brought Dog to your Garrison, and then brought Dog to Legion Dalaran. You can check to see if this was the case by going to legacy Legion Dalaran and seeing if Dog is in the pet shop on the character that you’re on.
If he is, you can go to the wreckage area even after you get out of it and move on to Dornogal. You’re looking for a dog bowl to click on which is pretty small and almost unnoticeable:
If he isn’t, then check your Garrison. If he’s there, you need to get a Pebble (off the AH is fine, it doesn’t get used by the questline and people typically sell them pretty cheap) to show to the Dog to get it to go to Legion Dalaran. That should be fine to get the dog bowl to show up!