Thanks everyone for the responses and advice. I'll start him out by testing the chicken allergy first. He started having runny poops again this morning and late last night, so the Wilderness food may not be agreeing with him as much as I thought.
I can't find any LBP chicken-free, grain-free food, so I'm looking at the adult food options now. It's not my first choice but it would be better than him itching and scratching all the time. I've heard the best things about Orijen foods, especially Regional Red, but that is nightmarishly expensive. I did find a Blue Basics food that is chicken and grain free, but that didn't get a great score on Dog Food Advisor, so I'd rather look for other options first. There's a natural/healthy pet food store near me, so I will go in there today and see what chicken-free, grain-free options they have. Calcium levels seem to be the biggest issue with looking for food for puppies, so if I can't find anything that is good-quality and reasonably priced, I may end up having to feed raw and get high-quality kibble as a backup. It's not my first option but I already have dead rats in my freezer for my snake so stocking some extra meat for my dog won't be the end of the world.
At this rate, my dog, cat, and snake are all eating better-quality (and more expensive) food than me. Oh well, it's better than having sick animals or dealing with health issues later in life.