While I’m searching for Gose and Berliner Weisse in old Books I’m finding also references to other beer styles that I really like, one of them is Lichtenhainer. In a nice book called “Enzyklopädie der technischen Chemie” from 1915 I found some nice info about Lichtenhainer. (This info & data is based on analyses and observations around 1910 so it could all been different 100 years ago)
Lichtenhainer beer is a slight sour, pale beer tasting smokey. It is similar to the Berliner Weisse but has a low level of carbonation. It is produced from smoked barley malt and gets is slightly sour taste not from the main fermentation but develops afterwards through an infection with lactic acid bacteria. These Bacteria are able to grow in the lightly hopped and shortly cooked wort. The beer is turbid most of the time and is drunken fresh before it develops a real sour character. The beer ist mostly served from cask.
And there is some analysis of the beer too.
The Lichtenhainer beer is a slightly sour pale beer. But it has a special taste due to the usage or partly usage of smoked malt. In difference to the Berliner Weissbier it is produced only with Barley Malt. When creating the beer the producers deviate from the old method in order to get better, clearer and more stable beer. So the deviated instead of using unboiled wort to use cooked wort but with a really short boil time. Not much hops are used in order to ensure the souring of the beer or not to delay it. …
He also says it was a drunk by students but nowadays (~1930) it fell out of style.
I would suggest following Recipe:
60% Smoked Barley Malt
40% Pilsner or an even paler floor malted barley
7.5 g per kg malt of Spalter Select (5 gallon batch around 20g) which you mash hop or boil on the side with a small portion of wort
No boil or short boil what you prefer
For Yeast either your own Berliner Weisse or a mix of German Ale (Wyeast 1007) Lacto Brevis (WLP672) and Brett Brux (WLP650) I also think a sour kveik culture like Hornindals would be kind of authentic.