Some experimental methods that solve the "umbrella blocks air flow" problem.
1. Set out two identical bucket-and-thermometer setups, placed so that a nearby object (a fence, billboard, or sign for example) will cast a moon shadow across only one of them. The object should not be a building or a cinder block, brick, or stone wall, because these retain heat and retransmit it at night, which will skew the results. Collect temperature data for several nights before, during, and after the full moon (these are your Experimental data points). Do the same thing with the same setup on several nights before, during, and after the new moon (these are your Control data points). If the Control data shows consistent results between the two buckets, your setup is free of external temperature effects and your Experimental data can be used.
2. Set out two or more identical bucket-and-thermometer setups, placed so that a nearby object (a fence, billboard, or sign for example) will cast a moon shadow across one of them for part of the night, and across the other one for a different part of the night. Collect temperature data for several nights before, during, and after the full moon. If the moonlight is cooling the buckets (in addition to the effects of the lower atmospheric temperatures that are also cooling the buckets), you should see the rate of cooling drop off during the shadowed part of the night, then ramp up when the moonlight once again shines on the bucket. This rate change will happen at different times for bucket A and bucket B.