We had same problem with our bulldog, we was told shampoo with hibiscrub absolutely brilliant his coat is beautiful now, before it was dry scaley and bald patches , you can buy from chemist,Wow! He is a good looking bulldog. That sounds like allergies to me. Welcome to EBN!
How often are you bathing him? And what is your rope cleaning regimen like?
There are some very knowledgeable members on this forum that may be able to make suggestions based on the ingredient lists for the foods you are feeding. To help them here they are:
Salmon and Sweet Potato:
Lamb and Oatmeal