That “screw” is commonly referred to as a “breech pin”.
I wonder why this gunsmith didn’t recommend using a drop of blue loctite? That would probably be the first thing I’d try. The piece of wood or shim recommendation seems silly to me, because that pin threads into the trigger plate, the threads are at the bottom of the pin. That shimming method works well with a wood screw for tightening things up, but not for a machine screw.
Try a drop of loctite. If that doesn’t do the trick, look into having another breech pin fitted up.
Btw…tighten the pin to where it’s actually tight. Not where it’s “timed” correctly. If you want a pin that is both tight and timed correctly, then have a new one turned and fitted.
I recall a gunsmith (I think Hugh Lomas) telling me that tight is more important than in proper alignment.