Yes, my son was diagnosed with RAD when he was under 2. He is 8 now and carries a diagnosis of asthma, which is now well-controlled. He had the same breathing problems as your son up until age 6.5 or so.
We are loaded with food and environmental allergies in my family, myself included, so I knew this was a possibility for him. As it turned out, the allergist did food testing earlier than usual with him because he was having so many problems. At the time, he was allergic to all dairy, soy, egg, peanuts, tree nuts, and other more rare food allergies as well as cats and dogs.
Not having dairy in his diet didn't result in anything different in terms of reducing the symptoms whenever he had a cold (labored breathing, intense and prolonged coughing, wheezing). Every fall, he was on the nebulizer from late Sept. or early Oct until summer basically. Once he hit age 7, that seemed to lessen, and for the past two winter seasons (this one and last), he has not had to be on the nebulizer continuously. I don't even think we used it this year, and he has a cold now.
He outgrew most of his food allergies except tree nuts and sesame and the environmental allergies. He still has a very strong reaction to tree nuts and ended up in the pediatric ER last year with an anaphylactic reaction. Very scary.
You're right that there are connections between RAD, asthma, and food allergies. There are also connections between these and GERD and some autoimmune disorders. However, each individual's specific connection varies; not all patients will have all of these diseases or disorders. It takes time, a great diagnostic specialist, and monitoring to find out what the particular triggers and disorders are for each patient.
I'm not against natural approaches, and it certainly won't hurt your son to eliminate dairy from his diet to see if it helps, but with things like asthma and allergies, it can be very dangerous and life-threatening if breathing and systemic symptoms are not addressed quickly by standard medically accepted approaches.
For my family, it's just not something I'd take a risk on, so I found the best allergist in our area, and despite some challenging times with the allergies and asthma, it was the best decision for us. The allergist helped us (my son and I are both patients) and improved my son's safety and our quality of life tremendously. I don't think you can go wrong by insisting on your son seeing a specialist in this case.
Best to you and your family.
J. F.