While the Docs aren't trying to worry you, I would definitely address this issue with your son and stay on top of it, especially if your family has issues with weight. It sounds like you are doing all the right things by eliminating his weaknesses, monitoring how much, providing veggies and meat as well as watching sweets and calories from liquids. I would encourage you to put in a plan of exercise for him and reward with some "treats" but mostly fun things to do. You are absolutely right in that he needs to be in charge of this too. Find something that he enjoys and try to get him (and you) some exercise every day.
My daughter was also born with food allergies, wheat being one of them. In learning more about her, I am learning more about myself. I have struggled with weight all of my life and my daughter is/ was on the chunky side and I too attributed it to my daughter's allergies. As a now diabetic (every women in my family has this), I now feel like food allergies have a played a huge roll in my diet. I am now of the mindset that we eat more of what we are allergic to and it's because our body doesn't process it right. For me, refined sugar is my "allergy". Once I took myself off of it, I felt so much better. My skin was better and I actually started losing some weight for the first time in a long time. Anyway, my point is that your son might still have some allergic reations to these foods and while they aren't as severe, his body might not be processing it correctly thus causing him to eat more of it and gain weight. Something to consider.
Hope this helps.