Heat remaining 2 tablespoons oil in a large nonstick skillet over medium-high. Add mushrooms and cook, stirring, until browned, 4 to 5 minutes. Add bell pepper, scallions, and remaining 1 teaspoon ginger and two-thirds garlic; cook, stirring often, until softened, 2 to 3 minutes. Add shrimp to skillet and cook, stirring occasionally, until pink, 2 to 3 minutes. Add bok choy and cook, covered, until bright green and tender, about 2 minutes.