This sounds just like Protein Losing Enteropathy to me....
Protein Losing Enteropathy in Dogs & Cats / Dogs Articles / Pet Articles / Televets.com
Has your vet run a TLI, folate or cobalamine test? The cobalamine is important as your pup may need to be getting B12 injections to help get back to health.
Regardless of the diagnosis, if getting them back to health requires a special diet, I would highly recommend consulting with a pet nutritionist rather than leaving it up to your vet as they do not have much information on nutrition other than the standard prescription diets. Special supplementation may also be recommended.
A defininite start would be adding a quality probiotic as this will help get the GI tract to optimal conditions.