Hitler did have a place on the Obersalzberg, but Himmler, Goering and others also had houses lower down on the mountain, and support troops had facilities in the area. I spent many nights in one of these that had been taken over by US forces, and renamed "General Walker Hotel". The Rod and Gun Clubs had the annual Convention and Trophy Show there. When you see the area and understand the joy of victorious troops, it is easy to understand how items coming from Himmler's or Goering's house might be remembered as having come from Hitler's. It is also easy to understand how items might be traded between the troops and precise location of capture not be too important at the time. Also, I'm not convinced the inscription is to a "sniper"(in our understanding), but rather to a good "shot" or "shooter".
Mike