OP,
You should be fine feeding the puppy food. One of the issues with puppy food is a balance in minerals, especially calcium, if I recall. In adult, or “All Life Stages” dog food, the mineral balance is not SO critical as it is while the puppy’s bones, skeleton and body are maturing.
It can be so frustrating when trying to transition to any new food, if you’ve found one your pup has done well on. But it can be a good idea to find more than one food your dog can eat. Recipes and even manufacturers can change, lines of pet foods are discontinued, or plants may shut down for whatever reason.
One of the things I have learned over my “dog years” is to vary what I feed my dogs. I started doing that after I had a dog with true inflammatory bowel disease, diagnosed by biopsies. All was well until the one food she could tolerate had a change in ingredients. 
I was stuck trying to find a quality food that she could eat but not have…adverse consequences (to say nothing of her discomfort, and painful episodes). One of her triggers was any chicken product (except eggs), and trying to find a commercial food that didn’t contain ANY chicken, even chicken fat, was very difficult, especially some 15-20 years ago.
Early on, I found that when I fed my pup a single puppy food over months, then making any changes in the diet could result in digestive upset. But now, if I vary what I feed the youngsters, I have less trouble with sensitive GI tracts or the unpleasant results. My dogs can eat a wider variety of foods, easily, so I don’t panic if I am running low on kibble, because I have other options.
So even as pups, I feed fresh food (cottage cheese, yogurt, eggs, meats, small amounts of organs like chicken liver, even raw chicken wings), different kibbles, a variety of raw foods, convenient freeze dried and frozen options, and an assortment of proteins. These days we have many, many more options for quality kibbles than a decade ago.
You may find that you can vary some foods with your youngster, if you can sleuth out what the pup may be sensitive to. Sometimes feeding a similar food, from a different maker, can make a difference. Adding in just a touch of yogurt, steamed rice (or even the water from steamed rice), cooked lean hamburger, and things like that, can introduce your pup’s gut to a wider variety of foods. What many people think of as “allergies” may actually be sensitivities. Or the dog may simply not have experience with a given food or ingredient.
Just like our gut, and like our horses, dogs have bacteria in the gut that help digest and utilize food. A pre/probiotic supplement can help ‘seed’ the gut with good bugs, so it can manage and use a wider variety of foods.
There are a multitude of books on feeding dogs, some of which are outdated in their information. One that I recommend when asked is “Raw and Natural Nutrition for Dogs”, by Lew Olson. Don’t be put off by the “Raw” in the title; the book covers all sorts of food types. The book was updated in 2015.
The Whole Dog Journal also has lots of information on feeding, including annual reviews of kibble/raw/frozen/freeze dried, and articles on dog diets and nutrition.
