A 7-year-old boy presents to the emergency department 2 hours after a fall from a height while playing. He has severe pain and swelling in the right elbow with the forearm held in flexion. X-ray confirms a displaced supracondylar fracture of the humerus (Gartland Type III). Neurovascular examination reveals intact radial and ulnar pulses, normal capillary refill, and normal motor and sensory function. What is the most appropriate immediate next step in management?