Summary: connecting the chamber to the ground did not have any apparent effect on the wandering peak.
We connected the vacuum chamber to the ground using a power cable connected to the rack. See pictures for details.
Before connecting the ground pin directly to the chamber, I used a multimeter to measure the voltage difference to be 17 mV rms.
After connecting the chamber to the power cable, and before connecting the cable to the rack, I measured the resistance between the chamber and the ground pin of the connector which plugs directly into the rack. Such a resistance was 0.2 Ohm (10 m cable plus and extension).
After connecting the cable to the rack, I measured the amplitude spectral density of F1 LVDT. The wandering noise was still there. See the last screenshot. Connecting the chamber to the ground did not have any apparent effect on the wandering peak.