My DS would have eaten several packets of crackers and half a jar of PB at that age, and he would have followed that by two full helpings of dinner, followed by eating any leftover dinner about 10pm. I never had to put any leftovers in the fridge for the six years between 11 and 17.
DS is allergic to dairy so he regularly drained whatever dairy alternative with added calcium/vitamins I had bought (whatever was reduced/on sale was what he drank on any given week). Then he would have put the empty carton back in the fridge but that is another story
In short, you are being VVVVVV Unreasonable, and I urge you to get a grip and also to buy more substantial snacks for your DS. His growth will be stunted unless he is allowed to eat as much healthy food as he wants at this stage.
Suggestions for healthy snacks:
Hard boiled eggs
Deli meats - ham, turkey, pastrami, etc
Whole grain baked goods (rolls, baps, buns, whatever they are known as)
Peanut butter, almond butter, cashew butter, other nut butters
Cheese
Veggies such as tomatoes, cucumber, leaf spinach to put in a sandwich
Good quality hot dogs or other sausages
Tuna salad fixings - mayonnaise, lemon juice, chopped celery, grated carrot
Leftover dinner that he can reheat
He needs lots of protein, calcium and healthy fats at this age.