You could try a new RCD, I'm not sure if they have higher rated versions like they do MCB but you could ask. We often have trouble with In-rush tripping MCB's when using VFD's but we just change them out for a "D" rated type and it fixes the issue. I suspect you could have something similar going on but with RCD.?

After that then it very much sounds to me like the spindle is shorting to ground and that's why it's tripping the Residual current device. You have Earth leakage and if it doesn't happen when the spindle isn't grounded then to me it must be the spindle.

Edit: I use have an ABB VFD which as a Screw in the side for EMC filter grounding and when fitted it does the same and trips the RCD. Does it have anything like this.?