You can begin the elimination diet by putting him on the hydrolysed protein prescription food that your vet has mentioned. If there is a wet food version, choose that just in case your dog has issues with storage mites. These are associated with dry food. The idea is that you feed this and nothing else (including treats) until you see an improvement. You then add something else to it and monitor progress. If the dog doesn’t respond then you have found an ingredient that he can tolerate so you can then try something else. The whole process can take some time so be patient.
Commonly beef and chicken are proteins that some dogs have problems with so initially you might want to avoid those. That being said, almost anything can be troublesome. As an example, one of my dogs can cope with any protein but is intolerant of legumes and white potato.
We have some information about veterinary nutrition specialists here.
They work remotely so distance would not be a problem. However, your vet might know of someone more local to you. The downside is expense but it could be cost effective in the long run because it has the potential to reduce veterinary consultations. More importantly your dog would be in better health if they can locate the problems.
Raw food is frequently cited as being very helpful in skin conditions and it is something you may want to think about once your dog is improving and no longer requires Apoquel. As mentioned, I would recommend one of the reputable companies who specialise in raw feeding and sell completes.
One other option which could simplify things considerably is to contact Different Dog.
They are manufacturers of fresh cooked, high quality food. They have a range of products but a number of them are based on beef and chicken. However, there is fish, lamb and turkey options. You can speak to their vet who will give you advice. It is expensive but sometimes smaller feeding amounts can be given with high quality food.