I have done some research into this, but not made one.
Dogs skin has a different pH to humans which is why you should never use human shampoo on them. However, from what I understand dog shampoo has a pH similar to cold process soap, which means it should be suitable (in theory). I would stick to things like olive oil, lard, palm, and Shea butter. As dogs aren't meant to eat chocolate, I personally would avoid cocoa butter (I'm not sure what part of chocolate they react to. Safest IMO to avoid, especially as there will be residual cocoa butter in superfat).
A lot of dry, red, itchy skin on dogs can be due to allergies/sensitivities which is why I would use a small number of oils in the recipe. Maybe a Castile would be best. In the information I read some EOs are ok - but I would definitely encourage you do research this for yourself before using.
As I said, I have not made any soap for dogs, only "researched" it (via google), and the comments above are just my personal thoughts on it - definitely not speaking from experience.
One suggestion for soothing dogs skin is to make them a bath in colloidal oatmeal -
Oatmeal in sachet/bag thing, add dog and water and let doggy have a splash around for a while.
My dog hates water so there's no way this would work for him!