My method for removing the cock is to disassemble the lock until just the lock plate, tumbler, and cock remain. If you don’t disassemble locks often id recommend taking photos of the process to aid reassembly. Find a deep socket that the tumbler will fit inside without touching, but still supports the underside of the lock plate. You’ll need a squared head punch that fits the inside of the cock but still has the contact area to avoid thread damage to the tumbler. I make mine from mild steel 1/4” square stock. Place the socket on a vise pad or hard surface, place the lock plate and tumbler on top and give the tumbler fitting a good wack with the punch and a medium size hammer. Nothing violent, just a whack, and the tumbler will drop into the socket. Works every time and no damage.