If the sensor is truly can reach that high resolution and sensitivity, it's theoretically capable of detecting the gravitational effect of the Moon.
To test the theorem I made a Linear FCDT especially for this experiment. This sensor should have a big L/R ratio (L: length of one coil, R: radius of the cell) (whys explained in this log: https://hackaday.io/project/11225-a-new-high-accuracy-tilt-sensor/log/46957-theoretical-background-of-the-linear-fcdt), and extremely symmetrically winded secondary coils. I made a sensor that has an L/R ratio of 20 or so: it's 196 mm long, the cell's inner diameter is 6 mm, has 5 layers of 0.25 mm coil wire in each coil, and looks something like this:To get reliable data from the sensor, we have to take it to a very massive and vibration-free place and make measurements with it for at least 3 days in a row. We found the proper place at the University of Pannonia on top of a high-tech CNC machine. To make it convenient, I made a little program in LabVIEW that sends the measured data to my email address at every 12th hour for 3 days (the program's VI can be found in 'Files' section).
The measurement was started on 7th of Oct at 11:47 a.m. (GMT+2:00) and we're planning to continue it for 1,5 week.
We got some anomalies in the data, but there is one encouraging fact: we detected a little sine wave signal that got a periode of 24 hours.
This means that we almost detected the gravitational effect of our Moon!