i dont have personal experience, though diets high in protein can be a cause, and pellets are high in protein, so because theres that risk of that being the cause, it is best to limit the protein--meaning pellets, eggs, insects, meat, nuts, etc.
pellets are unfortunately one sized fits all and some species can have overdoses on some of the nutrients in pellets, particularly protein, vitamin A, and calcium, especially if they are a high percentage of the diet or are supplemented with other foods high in these things.
best way to provide a balanced diet in birds is to offer a seed based diet, supplemented with fresh veggies and fruit as well as a lower percentage of pellets. variety is the best. you also want to avoid diets high in iron (so not much pasta or rice) a bit of everything, but for now since the vet said to remove the pellets, stick to seed and a good strong variety of veggies.
if you can get sugar free cranberry juice and put it in his water until the water is light pink, this would be good, or offer fresh cranberries daily if he will eat them. cranberries are excellent for supporting kidney health, and they can do wonders!! a few breeders i know swear by them, and ive had UTI's and i drank cranberry juice by the gallon for a few days and it went away with the cranberry juice, so i personally believe it works. its well known for its health benefits for the urinary system, which the kidneys are a part of
ginger is also a good herb for the kidneys, you can buy pure powdered ginger at most stores, you can sprinkle it on his food, or again feed him fresh if he will eat it.
you dont want to totally eliminate the protein out of the diet, but you most definately want to reduce it.
i know diets too high in protein can cause gout, which is not a good thing.
good luck with your too, i hope all is okay!!
some links of interest for you