My kid has similar preferences (but fortunately none of the allergies!). I pack his lunch with a couple of flat freezer packs to keep it cool. I always include a drinkable yogurt (like this
https://www.stonyfield.com/products/lowfat-yogurt-smoothies-strawberry-12-ct/) & a cheese stick. Between those two items, he has 10 g of protein (he's four, so suggested protein intake is 19g/day).
I also bought a barrilla pasta that is high protein and has 10g per serving. I've made pesto pasta with peas and feta and its a hit. I have also used the high protein pasta to make him an Asian-inspired noodle salad that he goes wild for and requests in his lunch (shredded carrots, bell peppers, and a dressing of toasted sesame oil + rice wine vinegar + ginger + garlic + a little honey + a touch of soy/ aminos -- sprinkle with sesame seeds if you have them). It sounds like work but I typically have the ingredients on hand and make a big batch for a dinner side and then pack him leftovers. It saves well.