
I'd ask for a referral to a Dermatologist tbh.
A customer of mine had a Westie as a pup, and he started with immediate skin problems as soon as he'd had his first injections.
For years the dog put up with this problem. Her vet said he was 'just allergic to everything' and he rubbed his skin raw and he cried through the night with the pain, and after numerous attempts, she finally agreed to see one of our vets who specialises in skin.
Immediately he did skin scrapes and firstly he was found to have a very bad bacterial infection, probably caused by the scratching. Then he was found to have sarcoptic mange so he was started on a special shampoo wash and had antibiotics and injections to sort the bacterial infection and the mange.
He also did a full blood count and allergy tests and it turns out that he was allergic to house dust mites, storage mites, eggs (which would have been in his innoculations, hence why the problems started then) chicken, beef, rice and gluten. He strangely, wasn't allergic to anything like pollen and grass etc too. They stopped all his food, and biscuits etc and started him on just plain jacket potato and salmon.
He was supposed to have come from a reputable breeder but she spent thousands on him. It was a shame because once he was sorted out and his fur grew back, and he was on a good food that suited him, he was looking unrecognisable as the same dog, he was taken ill and he was found to be diabetic. Unfortunately our vet couldn't stabalise him as he was also allergic to some of the insulins and she subsequently had to have him PTS :( He was only 5.
So, if he were my dog, I would be asking my Vet for a York allergy test and probably skin scrape.