Trouble initiating I2C communication between Arduino and 3.3V Slave
Problem solved!

I tried to get the FDC working with an Arduino Mini Pro 3,3V and had the same problem as before.

In the end, the solution was simple: The FDC has a shutdown Pin (SD), which sets it into sleep when HIGH and sets it active when LOW. When I connected SD to GND, all problems went away. Now I have a stable signal between Arduino Mini Pro 3,3V and FDC2114. I think it would also work with the Arduino Mega and level shifter, but I can't test it out.

Thanks again for your help.