I was in a relative's house over Christmas and came across the hypo-allergenic shampoo that I'd noticed and tried before. I'd never been able to find anything like it in the shops and now I know why.
It's called Epi-soothe and when I read the small print I discovered it is prescribed by veterinarians for use on cats and dogs. I didn't try it again. However if it worked before there is no reason why it should not work again.
I guess it hasn't been tested to the same standards as shampoos sold for humans. If you are concerned about animal welfare, you may think that is a good thing.