I don't sell a dog shampoo bar (I think the chances of getting sued over someone's pet having a reaction are FAR higher than a soap making a person itchy; and with so many pets with allergies, that's taking a big risk), but make one for use on our dogs.
Mine is a regular CP recipe (using whatever oils I happen to have on-hand-- usually a blend of olive, lard, castor, coconut/PKOand soybean) and I add 1 oz of castor PPO without changing the lye calc. It makes a bubbly bar that is moisturizing and rinses well.
To mine, I generally add a bit of neem oil, honey, and a few drops of lavender and peppermint EO. In the winter, I'll put in a bit of finely-ground oatmeal.
My show dogs have great coats and the handmade soap doesn't soften the coat too much like many commercial shampoos do.
Some pets are VERY sensitive to EO's. Pennyroyal, Tea Tree, Citronella, and White Birch especially. We had a groomer in town offer a 'spa treatment' using a soap with tea tree oil and three pets ended up at the vet for ataxia and seizuring. I have NO IDEA how much was in that soap, but it was enough to really affect those little dogs. And if there is any chance a client would use a soap around a cat, I wouldn't use EO's at all. A tiny bit of FO is probably safer. So many pet soaps tout using 'natural essential oils' but pets seem to be far more sensitive to these than people, so use them with caution.