Both my kids were really young when diagnosed (<2) but my daughter especially was STARVING when eating gluten. Used to wake up at 3 am to have a full meal because she wasn't absorbing nutrients. Now that she is 2 1/2 years gluten free, she eats about normal. While his behavior maybe reinforced by his peers and his age probably contributes, I would bet the celiac is a strong factor
My son is sensitive to wheat/gluten and showed similar behavior to your son's. He also does this when he has a low blood sugar level ( he has diabetes too). I wonder if your son is experiencing episodes of low blood sugar from poor absorption due to celiac, and hence the "munchies" type behavior?
After going gluten free, my son's aggressive eating attitude, and constant hunger abated, and is rarely a problem now. His blood sugar profile has greatly improved too.