Final answer:
The best method for immobilizing a pelvic ring fracture in a patient with signs of shock is using a pelvic binder, which helps to stabilize the region and reduce hemorrhage.
Step-by-step explanation:
The best method of immobilizing a pelvic ring fracture with associated signs of shock is Option 3: Using a pelvic binder. Applying ice packs might help reduce swelling, but would not immobilize the fracture. Administering painkillers can help with the pain but also would not address immobilization, and performing gentle massage is contraindicated as it may cause further damage. A pelvic binder is designed to stabilize the pelvic region, decrease hemorrhage and maintain the structural integrity of the pelvic ring, which is crucial for a patient presenting with signs of shock likely due to internal bleeding or other complications caused by the fracture.