I have a Tsing 300 infrared thermometer that I use for both cooking and A/C analysis around the house. It's an inexpensive model but gets the job done. It reads a 1" spot at 1' - I took readings at this distance.
So I went outside on a heavily cloudy night (so there was absolutely no moonlight to magically cool things down) and started testing the temperature of various objects in various locations.
First object is a painted railing inside a stairwell. Still night air, all measurements taken on parts of the rail shaded from the two 60 watt bulbs that are pretty far away.
Bottom: 69 degrees
Middle: 70.7 degrees
Top: 75.1 degrees
So height seemed to matter a lot (for this spot, we can't assume this always holds), 6 degree increase over approximately 19.25 feet. So that's one possible source of error.
Down at ground level, far away from any building or lights I then took several measurements of the ground.
Open to the sky: 69.4
Just in shade of tree: 69.9
Ground closer to tree: 70.4
So NO moon light and it's warmer under the tree at night. But if I ONLY took these same measurements in moonlight I might incorrectly attribute this to the moonlight. That would be a FALSE conclusion.
Next, I walked around an area equally open to the sky and took ground temperature measurements over an approx 50' circumference, I got:
So every few feet I got different readings, up to a 5 degree swing, even a 4 degree swing in two spots not far from each other.
So this tells me that even very local conditions can vary greatly and magical 'lunar light cooling' has NOTHING WHAT-SO-EVER to do with it.
I was also able to easily reproduce it being warmer under a tree without any moonlight so it doesn't matter WHY it can be warmer under something than in the open air, we know that moonlight cannot be assumed to be the cause.
It could be biological activity, it could reduce the effects of wind, it could be higher humidity, or all of the above. Every spot is probably different.
In short, people claiming the moonlight has a cooling effect failed to have any control data.
This is why science exists - to study the ways we FAIL and make sure to avoid repeating those errors in the future. Sadly our Flat Earth friends eschew science and so repeat even the most gratuitous failings of our past.
This failure is the very tempting ut this experiment clearly shows no such thing.