First of all the folks at the shelter are not veterinary healthcare professionals. I would ask to speak to the vet that examined the dog and prescribed the therapeutic diet. Your request will probably be denied.
I would then request to take the dog to a vet of my choosing for an examination (you pay). The diet you mentioned is probably a food trial to rule out sensitivities so I would suspect that the dog may be suffering from allergies.
If this is the case the dog will need to be under the care of a veterinary dermatologist for the rest of it’s life.
There is no cure for allergies but there is effective treatment.
And, NO, there is no substitute for prescription dog food, the one you mentioned goes through a special process so that the dog will not respond to any of the ingredients. Hence, the price.
All commercial dog food is subject to cross contamination. If you think the cost of the dog food is too much I would not adopt this dog. Allergies usually require lifelong treatment, they have flareups so it can be expensive.
BTW: There are no veterinary healthcare professionals affiliated with this site. Some of the regulars give dangerous advice involving raw diets and such.
Good luck
