Ok so it seems that the spindle may be the problem, there is a different test for you to do not using the beep, you have to select Ohms to measure the spindle only
Measure with Ohms setting, at the Spindle Plug ( Power OFF )
Pin 1 to Pin 2=
Pin 2 to Pin 3=
Pin 3 to Pin 1=
Then each Pin to Earth which in you case will be the Spindle body, as your Pin 4 Earth is not connected
Yes that photo that I posted is for a different spindle, but may still apply to yours, so worth a try, but it most likely won't change anything
90 % of these spindles, the users have to fix the Earth connection in the spindle plug, you could of course, just to try use the Ground wire that you have going to the spindle, attach it to the spindle Body, ( Just for a Test )
Have you used shielded Cable from the VFD to the Spindle???