I may be missing something blindingly obvious here, but isn't there a second bronze piece below the one in your photo, seen on the underneath of the tailstock casting?
If so, surely tapping the end of the thread downwards will extract both stud and bronze part downwards.
The two brassy coloured bits should be free to move in the hole in order to clamp the tailstock barrel in place.
If it's a blind hole, so the parts won't punch through downwards, straight through the casting, then I guess you need to pull them upwards using a length of tube, a washer and a nut on the end of the stud. Since you've not got much thread to play with, you might need to pack it up several times.