Hubby and I made soap 15+ years ago. Before homemade soap really became popular. It was difficult to get even $2 a bar. And it wasn't cheap to make.
We did not formulate soap for dogs. However, we did make several recipes which some of our customers used on their dogs with good results. One contained oatmeal and the other had ground coffee (our Odor Eater bars). We made it for my hubbys dad who fished a lot and needed a soap to get the stink off.
I always enjoyed the chemistry of soapmaking, but it was an expensive hobby. We used a very simple base recipe: 30/30/30/10
30 percent of coconut, olive, and palm oil.
10 percent total of coco butter, shea, hemp, or jojoba. hemp was our favorite for soft skin - but super expensive.
1.5% vitamin E oil. (Only extra oil in batch.)
Enough lye to 100% saponification – no excess oil. Must be exactly weighed.
Our bars were hard, lasted an extremely long time without spoiling and lathered well. With the addition of the 10% ‘luxury’ ingredients they were also very easy on the skin. Any extra’s like oatmeal, coffee, were added last. For scents we always used essential oils rather than fragrance oils – as the essential oils were far stronger.
Hope my 02 cents helps.