Perhaps the first thing anyone should do before deciding to take a dog on a 2100 mile walk is to visit their Vet and get a medical opinion if the dog is up to the rigors of a long distance hike of magnitude. There are several places you will likely have to carry the dog up/down sheer drops and fording moving water are two of those occasions, so you should get your own MD to clear you for that added level of challenge beyond your own ability to make the attempt.
Until you get these medical opinions, everything else from no-dog area work arounds and carrying three times the normal home food amounts is speculation. With these opinions you can make a better assessment of the possibility and what you need to do for preparation and what kind of preparation the dog will need. Then dive into the logistics.