Shielding powerful HF EMI
This is a potentially complex problem which needs some expertise and patience to resolve.
The first thing is to characterise the source better. What is the inverter supposed to be delivering? A 380 kHz PRF (pulse repetition frequency) of 2-6 MHz damped sine pulses sounds unlikely, I'd hazard the MHz ringing is due to the unbalanced line termination. But is the PRF intended? What signal continues after the plasma is struck?
How long is the feed cable from the HT unit to the torch? If you think of it as a quarter-wave resonator, take the speed of signal propagation as 0.6 c (180 x 10^6 m/sec if I recall correctly, but do confirm that for yourself!), figure out the resonant frequency and look for that.
The key earthing point is the main earth point of the cutter supply power unit: everything needs to get back to that as directly and with as low an impedance as possible. This will minimise stray current loops and hence magnetic induction.
Try clamping an extra earthing strap between the end of the torch feed earth shroud and the workpiece (note the games to make sure it does not stretch to its limit during the cutting!).
For the 380 kHz the earthing system is moot. A star earth might be OK, or a ground plane. But if the 2-6 MHz persists - and frequencies well above that will also be emitted by the plasma - then a ground plane approach is essential. Basically, you go do indeed go mad with shielding, and every signal line must be accompanied by its return line, the pair shielded and grounded at both ends. Especially, avoid passing any lines through the loop created by the HT supply, torch feed cable, torch, workpiece and grounding cable.
You can do a lot of shielding trials using cooking foil and sticky tape. The classic technique with shielding is to wrap everything in sight with silver foil and ferrite clamps, until the interference stops. Then remove each bit of the shielding in turn to establish which bits are actually needed. If a removal exposes interference, put it back and move on to the next bit.
Two useful probes for leakage sites, which can be fitted in place of the usual 'scope probe, are a short bipole (two stiff wires bent in opposite directions, one on the shield and the other on the signal line, or else a small loop connected across the two. These will highlight gaps in the shielding but not the &quot;source&quot; of the EMI as such.
The enemy of shielding is poor joints. The EMI does not leak through the gaps, as commonly imagined, but the gaps create an impedance in the current path, turning the less-well-grounded piece of shielding into an antenna. This antenna then absorbs RF from the source side and re-radiates it with prejudice on the other side. Fixing the gap works by fixing the antenna effect.
The good effect of the ferrite ring on the grounding wire suggests that you have a current loop in the grounding, with something unbalanced running through it - effectively an air-cored transformer coupling energy between the two circuits.
Hope that's useful.