What reason does the doctor give for restricting carbs? Are you supposed to restrict even whole, unprocessed, boiled grains? That is highly unusual. Or is it simply that he eats a whole lot of white bread and it's time to cut back a bit?
Until I read your additional details, I was going to suggest hummus! What doesn't he like about hummus? You can make your own dips, and send in with lots of veggies to dip! Baba ghanouj is also awesome - roasted eggplant dip. Dip your veggies in a dip made from veggies!
Ezekiel bread is delicious, high in fiber and protein. It does contain gluten, so if gluten is the problem, then choose differently. But it is less processed and healthier than other breads, and made from sprouted grains and legumes.
Many schools forbid peanut butter, but sunflower seed butter is delicious. It is good in ants on a log: fill celery with nut/seed butter. Top with a few raisins. (Or Craisins, for ladybugs on a log!)
Does your son eat eggs? I notice you said his diet was mostly vegetarian, though yours is vegan. If he eats eggs, boiled eggs are easy and kid-friendly. And they can make a salad easier to present. If he doesn't eat eggs, try different presentations of beans. Toss beans in salad dressing to make them more exciting.
Salads are always good - sneak in extra nutrition with some seeds or nuts, depending on what the school allows. Pack vinaigrette dressing on the side, so greens don't wilt.
Fake hot dogs are a "sometimes food" - highly processed and not very nutritious. Please get him to eat some real, unprocessed protein foods like lentils, chickpeas, beans, and seeds. Try homemade bean soups or lentil soup - he may like those better than other kinds of bean foods. Tempeh is also great for variety, but of course soy foods are not for every day!
Quinoa is a superfood and a pseudograin - a good supplement for protein and fiber. Not technically a grain, and higher in protein than most foods we eat as grains. And it's gluten free! And totally unprocessed, unlike pastas and breads. Just whole seeds, washed, then boiled in water. Or boil them in veg broth, or even in nut milk or a mix of nut milk with water (add cinnamon and top with fruit and pecans for breakfast!)