Place rice in a bowl with water to cover; soak for about 1 hour. Drain.
Combine lamb, 1 can mushroom soup, 1 can tomatoes, red bell peppers, rice, onion, lemon juice, lime juice, garlic, salt, and pepper in a bowl. Mix the filling together with clean hands.
Line the bottom of a 4-quart pot with a single layer of grape leaves.
Place the remaining grape leaves on a flat work surface with smooth side down (ribs of leaves up). Place a rounded tablespoon of filling in the center of each grape leaf. Fold bottom sections of leaf over mixture, fold over sides, and roll toward the top of the leaf into a firm cylinder. Pack stuffed grape leaves tightly in the pot. Pour the remaining 2 cans mushroom soup, 1 can tomatoes, and any remaining filling mixture on top. Cover with a small plate to hold the leaves together.
Cook, covered, on low, until rice is tender and lamb is no longer pink, about 1 hour. Remove from heat; let stand for 20 minutes.