Thank you for the replies.
Rocky1 - I live in the UK - we do not have rabies here and there is no requirement for rabies vaccination unless you are taking a dog out of the country or returning from a country with rabies.
The dog has had serology tests for allergies as, according to the vet, the rash is typical of atopic dermatitis and responded well to a course of steroids and then subsequently a course of apoquel.
The serology test results were a strong positive for milk, positives for 2 types of storage mites and a borderline for dust mites.
The dog has always been fed on a dry diet and the vet advised that we discard the old kibble (which was done) and avoid any food with grain as storage mites can be present in any grain or cereal based food and the dog will react to any parts of the mites present in the food, not just live mites. He recommended a grain free wet diet or raw diet. I have only ever bought small bags of food at a time, so no, the bag was not open for a long time. Mammals sensitised to dust mites may also cross-react to storage mites and vice versa.
No, I have not bathed my dog. He is not used to it and I suspect would be quite resistant to bathing although it may be something else to try.
There are no fleas - this has been checked for and the dog is regularly treated.
Both my Basenjis have had new bedding since the older ones diagnosis, blankets are washed on high temp and tumble dried, we have hard floors throughout the downstairs but rugs, curtains and upholstery has been steam cleaned weekly, since the diagnosis. Vacuuming has been done thoroughly, including curtains and upholstery, and is done every couple of days.