Sauté onion, garlic, and mushrooms in olive oil in large saucepan until tender, 5 to 8 minutes. Stir in rice and cook 2 to 3 minutes.
Heat broth to boiling in medium saucepan; reduce heat to low. Add 1 cup broth to rice and cook, stirring constantly, until broth is absorbed, 1 to 2 minutes. Slowly add 2 cups broth and simmer, stirring until broth is absorbed.
Add shrimp, snap peas, and remaining broth to saucepan. Cook, stirring frequently, until rice is just tender and liquid absorbed, 5 to 10 minutes. Add beans and tomatoes; cook 2 to 3 minutes longer. Stir in cheese; season to taste with salt and pepper.