Maybe someone out there who is a clever machinist can work this out.

6mm is 0.236 inches. A 6mm screw turns in just a bit, implying that the hole size is close. Perhaps the diameter is right but the pitch is wrong.

[Link: en.wikipedia.org] lists a pitch of 1.0 and a pitch of 0.75 for 6mm screws. Maybe the fine pitch screw is what's needed. My search just now did not find anyone listing that pitch, but it might be out there.

In case American fasteners might work, a couple of rare screw sizes come to mind. A #12 (OD 0.2158") or a#14 (OD 0.2421") screw might just fit.