Look, I'm not a vet. I don't play one on TV and I haven't stayed at a Holiday Inn Express (is that a national campaign? If not, that's going to seem like a really odd comment), but I do know that for the better part of our first year I constantly battled skin issues and spent a fortune at the vet's. The first August we had a rash that looked very much like what Dougie has, and it even turned black. The vet treated with antibiotics and medicated shampoos, but it kept coming back. I'm not faulting the vet, but the vet was doing what he was trained to do - treat the symptoms, and not worry about finding the underlying cause. So I spent a ridiculous number of hours researching her symptoms and came to the conclusion that she was suffering from a systemic yeast infection that was aggravated by her diet. She was on all sorts of top notch, grain-free foods, but the rash continued to come back until we made permanent changes.
I cannot say with absolute certainty that our situations are the same, but I will say that the food you're feeding has a bunch of the same triggers that set us off: chicken, eggs, cheese, and lamb all set us off. This is going to sound insane, but have you looked at a food energetics chart? Once I started following that for protein sources, our issues stopped. Take a look at this chart
here and look at the neutral and cooling proteins. If it were me, I would eliminate anything but proteins from his diet for a month and see what happens. Yeast feeds on the sugars from starches in foods. Any fruit, cheese, starchy veggie, and the binding agent of kibble will be loaded with carbs which will feed the yeast. Vital Essentials is a line of foods that gets 5 stars on dogfoodadvisor and all varieties contain only meat. I'd try their duck niblets and see if you're still having issues. It's a good cooling protein and is generally tolerated well. It'll also have a good balance of fat to protein for an active puppy.
Give him a bath once a week and then spray with that antifungal spray every night. I only had to do it for a week and there were immediate results. I give Beezy the equivalent of 2-3 TBSP of kefir every morning and she tolerates it well. I also give her a good fish oil in the morning and some coconut oil at night. Coconut oil has antifungal properties. Quinoa also has antifungal properties, so when I recently had a yeast outbreak I fed her an ounce of that with each meal and it seemed to help.
It sounds overwhelming, but you'll get through it. We're all here for you!