Is your vet sure this is a food issue? It is highly unusal for a puppy to react to so many things..and for it to happen to both. Have parasitic, infectious, and genetic issues been ruled out?
If it truly is the food (which I highly doubt), then I would be trying an rx diet at this point. What one you try is something that your vet or a veterinary nutritionist should decide. Most wouldn't be suited for long term use in puppies. and also, HA not having any nutritional value is not at all true.
I'd be getting a second opinion because this sounds more like a disease process and not a food intolerance.
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