Here's another thought; there's a bold "S" stamped above the 8,7mm stamp on the rifled barrel of your Drilling and perhaps this is what led someone to speculate it was made by Sauer & Sohn. Now they may be right, but I also have a pre-1939 Drilling marked Ernst Steigleder on the barrel topside, but hidden inside the foreend is "SuS", meaning Sauer und Sohn. If you look at the early Sauer factory catalogs, this was always their trademark and they had a close association with Steigleder. So I'm thinking that your bold "S" is something else again, (tempting to say Suhl but that's probably just another red herring).

Also, it seems to me that, the eagle proof mark varies in style when looked at closely from one German proof house to another and someone might recognize if that was the Zella-Mehlis one or not on your firearm. It might be useful for someone to photograph & catalog the different types for future use.