I think we are discussing different things...
Flat Earth Logic says that the Sunlight Warms things and the Moonlight Cools things...
So, conventional wisdom is that if you put two 5 gallon buckets of water side by side and block the sunlight from hitting one bucket, it will be cooler than the bucket of water in direct sunlight.
What Flat Earth Logic claims, is that if you repeat this at night, the bucket that is in Direct Moonlight will be cooler than the bucket of water right next to it...
.... How are we talking about sealed containers of food in the fridge? How are we talking about radiation escaping into the atmosphere?
Flat Earth Logic says that Moonlight has different properties than Sunlight, most importantly... that it actively cools objects in contact with it...
Silly Inflatearth wants to send me youtube videos as proof of this... But I have youtube videos as well... lets test it in the back yard....
How are we talking about food in a refrigerator? Jesus Christ on a Cross gentlemen... I miss Jane... where is she...
Sealed containers in a fridge is a bit of a seg-way, but the idea is the same.
The night is cold, typically much colder than the water, meaning it would tend to lose energy, i.e. cool down, just like a container in a fridge.
If you put things in the way, such as an umbrella or a lid, you stop various forms of cooling. I just focused on radiative cooling, but there is also the evaporation of the water which the umbrella will hinder to some extent, and a lid certainly would, and convective cooling which again, the umbrella would disrupt and a lid would stop.
So the point is merely putting a bucket of water out an night will result in it cooling down, regardless of if the moon is there or not.
Putting things in the way, even if done to "stop the moon light" will result in that cooling effect being diminished.
Thus if you compare a bucket in the shade to one in direct moonlight with no obstructions, the one in moonlight will get colder. But the same happens if you remove the moonlight (from it being a moonless night) and instead just have a shaded bucket and an exposed bucket.
This is not the same with the day, as now the equilibrium temperature is much hotter and the water is typically cooler.
If you like, you can try the same in the day by getting a piece of red-hot iron and placing one in the shade and one in the sun, and seeing which is cooler. That would have a similar effect, but the radiation from the sun still might win.