Hello and welcome to the forum. From what you have said, the indications are that your dog might benefit from a change in diet - link.
There is some information about feeding dogs with intolerances and allergies here.
Re the prescription diet - I can appreciate your concern. David has written about prescription diets here.
I have mixed feelings about how to proceed because it is clear that your dog’s current food is not suiting him and the Hill’s might be useful in the short term. If it alleviates the symptoms, all well and good because it will be helpful for your dog and give you a base from which to work when looking for something better. I will leave you to think about that but would suggest that you don’t need to view it as a long term food.
On the aforementioned Feeding Guide
page is a section about elimination diets. This could be helpful for you to find out if your dog has any particular intolerances. At the outset, you need to keep it very simple indeed so a wet food would be the best choice. No mixing with other foods and no treats or scraps.
I am wondering if using fish would be a good starting point. There are a few simple wet foods, usually with potato as the carbohydrate source. Forthglade sell one with brown rice. Naturediet wet food is simple in recipe and they have a fish version. They had some useful information sheets on their website about this kind of issue - don’t know if they’re still on the website. There are some kibble type dry foods that are fish based and have fairly simple recipes. Of the cold pressed dry foods, Gentle Fish might suit but best to contact the company first.
Please let us know what you decide to do and if you need further help.