The short article in the IEEE Transactions on Nuclear Science (link, DOI: 10.1109/TNS.1972.4326697, S.K. Poultney "Single Photon Detection and Timing in the Lunar Laser Ranging Experiment", IEEE Transaction on Nuclear Science 19(3) 12-17 (1972)) in the answer to your companion question discusses the setup used by the author to make measurements at the McDonald Observatory.
Given the setup at the McDonald Observatory using the 272 cm reflector there, they expected to see 0.2 photoelectrons (signal) per laser firing. The authors have one paragraph labeled "Discrimination Against High Background Noise":
Operation under all background illumination conditions requires a variety of noise discrimination schemes. The extreme case is the detection of a single laser photon against the combined background of bright moon and bright sky. A spatial filter of 6 arc sec and a spectral filter of 0.7 A width still allow about 300k counts/sec of noise.$^{10}$ Another discrimination technique was the setting of a 6 microsecond gate about the expected time of arrival of the lunar return by electronic means. The final technique was the post detection clustering of returns within a few nsec. A number of repeated rangings were, of course, necessary to be certain of a signal return. The RCA C31000F (with ERMA photosurface) which is used as the photodetector exhibited a noise rate of 30k counts/sec at room temperature and several k counts/sec when cooled to about 0°C due partially to its operation at very high gain. The photodetector noise rate is only important when viewing the dark moon at night.
Endnote 10 is
S. K. Poultney, "The Detector Package for the Laser Ranging Experiment at McDonald Observatory: Its Design, Performance, and Operation," Dept. of Phys. and Astron. Tech. Rep. No. 957, University of Maryland, August 1969.
This shows up in theirthe university catalog, butand is not available onlineas a NASA technical report, along with others detailing the work under contract.
The impression I get is that the experiment was capable of operation regardless of background light ("Operation under all background illumination conditions") using good optical and electronic practices.