Hello and welcome to the forum. Although you don't mention vomiting, the symptoms that your Westie has sounds like
Bilious Vomiting Syndrome which is very common and occurs when the stomach has been empty for some hours. In many cases all that is needed is to reserve some of the dog's food to give immediately before bed. One of my dogs has this and tends to bring up a small amount of bile. I've found that giving
Dorwest Herbs Slippery Elm Tree Barks Powder before bed is very effective. This Easter weekend they are offering free p&p with use of code EASTER22 but IIRC it finishes tonight. I mix it with water and add some chicken. Slippery elm is a mucilage which soothes the stomach. You don't mention how often you feed your dog but a dog's stomach is very acid and in these cases smaller, more frequent meals might help. I divide the RDA between three meals for my two and then give a small amount of chicken and a biscuit at bedtime. That way, the stomach is not empty for long.
Regarding the dermatitis, I had Westies for some years and even then the breed was known for this skin problem. One of mine had it although I did manage to get it under control with Seleen shampoo but she did not have mites. Skin conditions are sadly quite common and it can be hard finding the cause. One of my dogs had a skin problem so I put together some information in
this thread. You might find something useful in there.
Are the mites properly cleared now? Has the vet done a skin scraping? I think that is the first thing to do. The vet will also be able to tell if your dog has malassezia (yeast). Both of these conditions need vigorous and sometimes prolonged treatment. All too often the courses are too short. Antibiotics are also sometimes needed because the dog introduces bacteria into the skin by scratching. Again, a longer course is often required. I find that a suitable medicated shampoo used regularly is extremely helpful.
Diet wise, the first thing you need to do is to look for food that is low in carbohydrate. This takes out any kind of kibble. there are a few that are average carb but most are very high - they need to be to form the biscuit. Low to average carb diets would be fresh cooked food, raw or some high quality wet foods. If you need help with sourcing something, please ask. A good diet is essential for skin health and the immune system. In the case of my own dog I've found that since she has been a low carb, fresh cooked diet her skin condition has remained in remission.
That's all for now but please post back if you need any help.