I think it's easier and more realistic to do an exclusion diet feeding home-cooked stuff when food is a big part of their lives, as you say. I put Luna on the kibble the vet recommended, but (as much as she ate it, because she's a Lab), I wasn't happy with it, and wanted to be able to give her more - and better.
When you home-cook, you want to ensure that, as well as having a proper nutritional balance, you feed a range of proteins, but for short-term, diagnostic purposes, it doesn't matter so much. With Squidge, I put her on a white fish and vegetables diet to start off with. I used a couple of different white fish (as I was pretty sure she wasn't sensitive to that), and a range of veggies. For training, I blitzed the food I made and used a squeeze tube. I would also make pyramid pan treats from the same stuff. Sometimes, I used Fish4Dogs white fish kibble.
In the end, the vet didn't think it was food at all, and put her on some meds, which seemed to clear it up, but then it re-emerged. There seems to be a correlation with lentils for her. Now, their regular food is a mixture of fish, pork and beef, with vegetables. I cook up offal separately, as Squidge can't have it, and I supplement with calcium and fish oil. They get raw chicken carcasses, or raw frozen fish at lunch, sometimes a chicken leg, or a portion of rabbit, or raw egg. They absolutely adore their food, and I have had no more skin issues.
If you're interested in home cooking, there's a very informative Facebook group specifically for it, here:
Home Cooked Diets For Dogs