Hi Jonas
The type of instrument you have was very popular in the late 19th- early 20th Centuries. It was called a hommel, or hummle, among many other names. In the French speaking areas it might be called eppinette. Check out Salon Ambrosine for a contemporary take on eppinette music from Wallonia.
In 1975 I met a Flemish musicologist named Wim Bosman at a festival in Ireland. He was traveling with his group, which featured four or five of these instruments. I believe he called them "vlier," although that may have been the name of the group. It was a long time ago. I seem to recall that some songs featured several instruments tuned to different pitches. So, for example, a pair would play in C when the song was in C, and a different pair would play in G for the dominant chord. I've also heard this technique with Hungarian citteras.
Mr. Bosman was then working at the musical instrument museum in Brussels. He was also involved in publishing an extensive survey of fretted zither instruments in Belgium and Holland. I have a copy of a short overview that he published in English; I'll see if I can dig it out of my files and post a pdf copy to this forum.
From what I can see of your instrument, I'd tune all the melody strings in unison, say to G so you could play in C major. Ditto the drones--they could be C or G. I think @NoterMan has that covered.
Stay tuned!