Another idea for you
Hi there - We adopted an itchy boxer last year who will be 4 in Feb. We have tried everything under the sun, and still he itches. Sometimes it gets better, then it returns. For most of our time with him he would get so sensitive, that he quivered when he was just touched. He didn't enjoy being loved on, and that was really hard for me!
We have been seeing a holistic vet for a few months because we didn't want to have him on Pred and other drugs for the rest of his life. His coat does seem to be improving, albeit slowly. I would second the plan to give your puppy probiotics and digestive enzymes. Coconut oil, too. If he likes the taste, use it for a treat - that's what we do, Nico loves the stuff. But here is something to consider, and I know it adds substantial costs if it turns out to be right: Take him completely off kibble. Just try it. We were giving Nico a half cup of kibble with his raw food to help defray the costs of the raw food. We tried every high-end limited ingredient kibble out there. HOWEVER. We decided one day to just stop the kibble to see what would happen. He still itches, BUT he is no longer sensitive to the touch. We can pet him and rub his belly and no more twitching! The other thing is that he rarely has gas any more. On the kibble, he was a fart monster, I mean, many times a day of very rank farts!
You could try canned food for a bit or raw, transition slowly, but I would consider at least trying it. You have nothing to lose, and if he stops chewing his paws, it may be worth it. If you have any questions, feel free to ask. Good luck!