When you put the screw in and do it up a small amount of glue is likely to ooze out potentially gluing the core to the shaft of the 3x3 middle. So put a tiny amount of vaseline around the screw hole (on the flat top) immediately after putting in the glue. This helps to stop these pieces bonding which obviously you don't want. It might still stick but it should be weak enough to break free by turning. Be careful because you need just the right amount of glue and vaseline to stick the screw securely but not the 3x3 middle to the core.
I can't guarantee it will work but I often do it on my large puzzles when there's a danger of screws coming loose and having no way to tighten them after construction.