This is an old revision of the document!
As I wrote in attempt number 3 at the end, this is the plan for attempt number 4:
There I will test out the bigger brother of the LDC1314: The LDC1614. It has the same pinout and overall package, so it will be an in-place replacement. The big advantage is the 28-bit resolution. Hopefully, there is no gain needed then.
I´ve got the needed parts some weeks ago (thanks Ferdi for ordering them!), but didn't have time to solder or even start programming because of my bachelor thesis.