While Hills Rx diets are life savers for dogs really needing prescription foods, you have a puppy and the need for ID seems rather extreme to me... and suspicious. If this is not a vet you simply adore, you might look around for one who doesn't jump to pushing rx food. I agree with Pat, Hills is simply nothing I'd use unless no other choice.
Second, I'd suggest getting off puppy food, period. Rather than going to an rx diet, getting the puppy on a no grain, low fiber should help. I love Fromm, but I never touch puppy food. With less bulk, the feces will be firmer and easier to push out. But walkers still walk, lol. It's just what they do, but hopefully less once the stool is firmer.