In reply to Fred and Okey:
I don't know what the Cicerone route for the Highlands section is, but the John o'Groats Trail from JoG to Inverness is a decent route. I haven't completed it all yet, still the most northern parts to do, but I can't think of anything in the sections I've done that would be difficult for a dog as long as they're on a lead when necessary.
If you're talking about taking the Cape Wrath trail potentially, so going from Fort William, the alternative is that the Great Glen Way from Fort William links up with the John o'Groats trail at Inverness; I think that would be easier terrain than the Cape Wrath trail, especially in bad weather. Easier to stock up on supplies on the way too so you don't need to carry all the dog food and your own food!