Preheat oven to 350°F; grease a 9x13-inch dish.
Melt butter in a large skillet over medium; sauté onion and pepper about 5 minutes until softened, then garlic 1 minute more.
Mix both soups, Rotel, sour cream, chili powder, and cumin in a large bowl; stir in vegetables.
Layer half the tortilla strips in dish.
Top with half the chicken, half the soup mixture, and 1½ cups cheddar.
Repeat with remaining tortillas, chicken, and soup mixture; top with remaining cheddar and all Monterey Jack.
Bake uncovered 40-45 minutes until bubbly and cheese is golden; rest 10 minutes before serving.