Heat 1½ tablespoons of the oil in a large skillet over medium-high heat. Add the bell peppers, garlic, and ¼ teaspoon each salt and black pepper. Cook, tossing frequently, until tender, 8 to 10 minutes.
Meanwhile, season the tilapia with ¼ teaspoon each salt and black pepper. Wrap each fillet with 1 slice of the prosciutto. Heat the remaining 1½ tablespoons of oil in a large nonstick skillet over medium-high heat. Cook in batches until cooked through, 2 to 3 minutes per side, adding more oil to the pan if necessary. Fold the basil into the peppers and serve with the tilapia.