Preheat oven to 375 degrees F (190 degrees C). Spray an 9x12-inch baking dish with cooking spray.
Rub the red bell peppers, yellow bell peppers, and orange bell peppers with barbeque rub; arrange in the baking dish.
Heat olive oil in a skillet over medium heat; cook and stir chicken, onion, green chile pepper, jalapeno pepper, turmeric, coriander, cumin, salt, and pepper until chicken and onion are lightly browned, about 5 minutes. Add beer to chicken mixture; reduce heat to low, cover skillet, and simmer until beer is reduced by half and chicken is cooked through, about 45 minutes.
Fill bell peppers with chicken mixture. Pour stewed tomatoes, tomato sauce, and chickpeas into the baking dish, around the bell peppers. Cover dish with aluminum foil.
Bake in the preheated oven for 45 minutes. Remove aluminum foil and add rice and Mexican cheese blend on top of the tomato mixture around the bell peppers; continue cooking until cheese is golden brown, about 15 minutes.