Hi Paul,
Take care when working on it while it is running... Personally, I wouldn't earth it, for fear of a feedback loop and blowing the AVR, but that does present the possibility of 'shock hazard' if you touch ground and gen frame at the same time.
In this case, I would use utmost care, de-mount the AVR, install the small globe between N-E, then start it up... if the power fluctuates as before, shut it down immediately... Without touching metal on the gen if at all possible.. Dry leather glove or a good rubber glove recommended... The chances of killing yourself are slim, but hey, why take a chance if you are not familiar with the angry pixies...they might try and bite you!
If the power fluctuates with the AVR unbolted from the plate, I think you will need a pro to come take a gander... If the voltage is stable, we have found the problem....
Also, when you unmount the AVR for the test, stick it in a plastic container or something to insulate it and prevent it from touching other metalwork and shorting out on something else...
Further, if this solves the problem, don't expect the 'AVR in a plastic box trick' to be a solution long term... it will need proper mounting without being enclosed for cooling...
Cheers
Ed