[emeraldecho]

With no improvement between the iodine and the medicated shampoo, it's unlikely to be bacterial or fungal in origin unless you've got repeated exposures to something in the environment. So I'd agree with the allergy hypothesis.

Allergy testing would probably be the quickest route to confirm, though if cost is a concern you can also try manually wiping down the paws and snout after going outside as @storyist suggested, and doing a limited diet (usually rice and a single protein source to begin) that gradually reintroduces other ingredients to test if they are the source of the problem.

I'd also caution that sometimes the ingredients in medicated shampoos can themselves be allergens. In your shoes, I'd probably use a salt water wash to prevent infection instead until you identify the allergen.