First I’d make sure the dog wasn’t getting any chemical flea and worming treatments, being over vaccinated or drinking tap water and the owner isn’t using lots of chemicals around the home and garden, all of these factors can effect the microbiota which could then lead to the dog having a more sensitive gut.
Initially I would recommend doing some work on the gut before even considering changing the diet, my go to would be a good quality soil based probiotic and slippery elm.
Generally starting with a light protein like turkey, chicken or rabbit is best but finding out what proteins, if any, that the owners know their dog is ok with would be the best to use initially. These are known as “safe proteins”. I’d start adding in small amounts of safe proteins, very lightly cooked and one at a time to their current food, if the dog is known to be ok with certain veggies they can be added too. If this is tolerated well, I would then start to decrease the current food and increase the fresh cooked food over a period of time at a pace that suits the individual dog.