In order for me to believe that moonlight is harmful, a test would have to be done under very controlled conditions. For example, to eliminate any environmental variables, I would measure the light spectrum emitted by the moon and then replicate it exactly using artificial lighting. I would use several variations of types of light exposure (sunlight & moonlight, sunlight and darkness, moonlight and darkness, moonlight only, sunlight only, darkness only), and I would at least perform the experiment on an animal vaguely related to humans. I would then develop some method to quantify the 'damage' caused specifically by the light. It would probably involve comparisons of overall health to that after an equal period of time exposed to normal lighting conditions and performing the same activities as during the tests.
It would all come down to eliminating the extraneous variables and quantifying damage, which none of the studies cited here have done.