I am thinking allergies too, My son has asthma and there was a kid in my other son's swimming class that always coughed all through class, it was allergies, and asthma that they had no idea.
I would def. still get the results, but I would have them run a full allergy panel on him and see an allerginist. This may be a food allergy where because you are stillintoducing the allergin daily he is reacting.
My middle child is allergic to eggs, I took him in to the Doctors. 7 days after he started with resperatory symptems, and the Doctors. to only hear from the Doctors that it is just a cold and needs to run its course. durring the next 7 days I sware his lips were turning blue but no one else saw it. took him in again only to hear it again its just a cold let it go. Now I am getting mad not 1 test, not 1 xray nothing, so in the course of another 7 days I found a different Doctors and made an appointment and now his nail beds were turning blue. THIS IS NOT IN MY HEAD. so the day before the new Doctors appointment we saw his regular Doctors and was told again that it was just a cold and I am waisting my time. I took him to the new Doctors and he rushed a xray blood test cultures, to origonally think it was pneumonia, to finally find out he is VERY allergic to eggs.
The point is not all doctors are the perfect fit for us and our families, if in a couple of days you do not like the route he is taking there is nothing wrong with seeking other opinions, it saved my son's life.
Good luck and please keep us posted on to his outcome.