Hello and welcome to the forum. I have recently compiled a thread on allergic dermatitis which might be helpful - link.
Have a look at that and please let me know if there is anything else you need to discuss.
I am not familiar with Chow-Chows but am wondering if the skin problems are genetic. Are they common in the breed? There is some mention of a genetic component in this thread on a Chow-Chow forum.
As the vet thinks that the skin problem is food related I can see why they have suggested the hypoallergenic food. It can be used as the base for an elimination diet and as your dog is so young, it is worth considering.
Long term, I think it might help to avoid high carbohydrate dry food. Alternatives would be fresh cooked food, raw or wet. If you are inclined to cook for your dog, this can be achieved safely but research is needed because certain additives are required, notably calcium and omega oils. We have information in the home cooking section.