Never tried that particular operation. I would expect to just push the wire in from the center/inner end and it would pop out the outer end. The tube is held in place with a couple setscrews. I'm not sure if there is a separate tube for each side or if one tube goes all the way from one side to the other. If separate tubes, removing them might help. If a single tube, it won't help.
You may have already tried this: Put a loop of wire up into the center of the tube. Shove another wire in one end through to the other, so the loop ends up looped around the wire. Use the straight wire to pull a string through the tube, having plenty of string sticking out each end of the tube. Use the loop to pull the center of the string out of the tube. Cut the string in the center. Use each half of the string to pull one headlight wire.