Heat olive oil in a Dutch over over medium-high heat. Add green bell pepper, red bell pepper, onion, and garlic; cook and stir until fragrant, about 1 minute. Add green chiles and jalapenos; cook and stir until starting to soften, 3 to 4 minutes. Add rice; cook until coated in oil, about 3 minutes. Add broth, tomatoes, Southwest seasoning, salt, and pepper. Stir well and bring to a boil. Add lime juice and stir well. Cover with lid.
Transfer to the preheated oven and bake while you prepare the meat, 10 to 15 minutes.
Heat a skillet over medium-high heat. Add pork and beef; saute until browned and crumbly, 5 to 7 minutes. Mix in taco seasoning, chili powder, and cumin; season with salt and pepper. Add black beans and stir.
Remove rice from the oven and mix in the meat mixture; sprinkle a few tablespoons of water on top. Continue baking until rice is tender yet firm to the bite, 20 to 25 minutes. Mix in green onions and cilantro; top casserole with Cheddar cheese and mozzarella cheese. Return to oven and bake until cheeses melt, about 5 minutes more.
