You have interesting problems and many helpful suggestions. So here are mine. Send the entire pistol + side plate + magazine to either of the members recommended. Have them not only fit side plate but also "tune up" the pistol.
In addition take clear pics of the front and back straps so that the police unit it was attached to can be ID. It's been mentioned that a 1936 police Luger is very rare so this would interesting too.
Aldo35
PS Keep looking you may find more gems!