You have my sympathy because I've been through the same thing with one of mine. Basically I went to a specialty pet store that has dozens of brands of kibble and went through every one looking for something that didn't have any of Story's triggers. And found only one limited ingredient food that came close - it only had one thing: alfalfa. Since alfalfa is used as a supplement, there wouldn't be a lot of it in the food, and I put her on that. The skin problems stopped, but her coat went to hell. Which is why I now feed raw.
So you'll have to do the same, go through every food you can reasonably acquire, read the ingredients, and try to find something that suits. You can research on the internet, but for once, I found going to that store and reading ingredients on sacks easier than online.
So raw fixed it for us, but I even had the unhappy experience of settling on a base diet for raw that made covering all nutritional needs easier. It was the only base diet that didn't have something she reacted to in it, and it was great for a year. Then all of a sudden she had the worst outbreak of skin problems ever. I suspect they changed their formula without notice. It could be she developed an intolerance to something that hadn't previously triggered her, but the outbreak was so sudden and so great I suspect a formula change.
Then there's the problem that she wasn't tested for everything in the world. I read about coconut oil, tried it, and bingo! Hot spot. She wasn't tested for coconut oil.