Well, my thoughts were the same as yours, about the ingredients, and Pumpkin reacted to even the stuff for sensitive coats.
The things that is used to make shampoos and soaps foam is lye. I believe my sweetpea is allergic to lye, because even all natural shampoos, with lye, made her skin scabby.
So, by definition, what I use on Pumpkin isn't a shampoo or soap, but emulsions, because I make them without lye.
I have my recipes written down on a piece of paper some where, so I cant give you the exact recipe right now, but what I use is two different emulsions, or "creams" with different properties, and a liquid to degrease.
I make a degreaser, with mint, tea tree and lemon to degrease the fur. There's a very, very small amount of lemon, so it wont affect the colour of the fur. Then I make an emulsion with coconut oil, mint and bergamot, because the degreasing liquid wont clean the fur, it will only pull out excess grease. The emulsion is rubbed into her fur, much like a shampoo, and it gets out all the icky stuff. The bergamot deep cleans, and the mint is there to cut the grease, so it's not as greasy as a conditioner. Then a conditioner emulsion with witchhazel, lavender,shea butter and coconut oil, this conditions and soothes the skin after the cleaning.
I use esential oils, organic coconut oil and shea butter, and mineral water. You can buy shampoo bases, and add the oils you want to that, but as my baby is allergic to lye, that isn't an option for us. For the emulsions I use a very basic lotion/conditioner recipe, as conditioner is basically a hairlotion. It works brilliantly for us
After Pumpkins next bath I will have to make a new batch, so I can film it and put on youtube if you like to see exactly what I do. I will also dig out the recipe