Cooked meat couldn’t hurt. Many owners that feed their dogs raw food generally do not have allergy issues! But, there is SO much that goes into feeding a raw diet, that it isn’t something I would reccomend unless you are prepared to research until your eyes falls out, get all the proper vitamins, and the list goes on and on with the things that you need to do.
Adding cooked meats can only help the diet of your dog, unless they are allergic to that meat! Ha ha! It seems to have helped my dog. Since you feed a fish diet, try to mix in canned sardines (in water and not oil) and canned salmon from time to time. They are packed with omegas that will help skin and coat as well. Eggs are a good source of biological protein for dogs that isn’t always obtained through kiln-fired kibble. It won’t hurt to try. Just do small amounts starting off, and gradually add more. Make sure that you reduce the amount of kibble you are giving in relation to the amount of meat or other item that you are giving. You don’t want to cause your dog to become obese!
Again, I would still contact the vet and see what they think about Benadryl. It is an inexpensive way to help combat the allergy symptoms. Many vets recommend it if the allergies are not severe, and many vets don’t approve of it. All you have to do is ask!