A 6-year-old boy presents to the emergency department 2 hours after a fall from a height onto an outstretched hand. Clinical examination reveals swelling and deformity of the right elbow with loss of normal carrying angle. Neurovascular examination shows absent radial pulse but normal motor and sensory function in the hand. X-ray confirms a displaced supracondylar fracture of the humerus (Gartland Type III). What is the most appropriate immediate next step in management?