Hello and welcome to the forum. It sounds like you are having a bad time with your puppy right now. You did the right thing by putting him on a better diet than the supermarket one. Exclusion diets have to be carried out very carefully and methodically, preferably with the support of a vet. Having just checked the formula of Barking Heads puppy food it looks quite suitable and as it is grain free it is a good choice. I assume it is this one:
Composition* Freshly Prepared Deboned Salmon 20%, Sweet Potato, Dried Salmon 16%, Freshly Prepared Deboned Trout 11%, Potato, Lentils, Peas, Sunflower Oil, Salmon Oil 3.5%, Pea Protein, Salmon Stock 2%, Lucerne, Seaweed, Prebiotic MOS, Prebiotic FOS Link
Analytical Constituents Crude Protein 28%, Fat Content 18%, Inorganic Matter 8.5%, Crude Fibre 2.5%, Moisture 8%, Calcium 1.8%, Phosphorus 1.4%, Omega-6 (4%), Omega-3 (2.7%), DHA 0.3%
Nutritional Additives (Per kg) Vitamin A 25,000 IU, Vitamin D3 2,222 IU, Vitamin E 694 IU
Trace Elements as Compounds Ferrous sulphate monohydrate 956mg, zinc sulphate monohydrate 772mg, manganous sulphate monohydrate 152mg, cupric sulphate pentahydrate 56mg, calcium iodate anhydrous 6.80mg, sodium selenite 0.77mg
Assuming that your dog hasn't got mites or has been bitten by anything then I am at a loss to know why he is scratching and so irritable. What you describe does sound like allergy but I take the point that your vet is making in that one week is not a long time to get established on a different food. Of course it is possible that his diet has nothing to do with it. There are dogs that are sensitive to pollen and there is a lot of it around right now. I recall that my youngest dog had a touch of hay fever when she was a puppy and it was this time of year.
You can give Chlorpheniramine (Piriton) to dogs but you really need to ask your vet first because yours is just a baby and you need to be sure it is safe and that you have the correct dose.
The other thing that might help is to clean your pup's feet and undercarriage when you come in from a walk. You can do this using plain warm water or purchase special wipes for dogs.
With the ears you can just use a small piece of damp cotton wool and wipe around the external ear twice a day (I assume your dog has dropped ears). However, I am just wondering if he has an ear infection as well - did the vet look down with an auroscope? I know that you have seen the vet but without knowing the full details, I think that it sounds as if some extra treatment might be necessary just to get him comfortable.
If you really feel that this is diet related then the only thing I can think of right now is to give him some home cooked food for the next few days. When mine are poorly I usually give well cooked brown rice, chicken and steamed white fish. It is not very well balanced but as a stop gap for a few days it seems to help.
Please let us know how you get on with him. I hope that he is soon better.