The effect of the moon on weather is indirect in that the moon affects ocean tides that can alter ocean currents that can affect weather. Also, the gravitational pull of the moon does produce atmospheric "tides"; however, the effect, while measurable is small and overwhelmed by other atmospheric factors. Another effect, that while known and measured for several decades at least, has recently begun to receive more attention by geologists is the tidal effect of land masses. In particular new satellites that can measure the topology of the earth show unmistakable "ups and downs" due to the gravitational pull of the moon. It has been hypothesized that these small shifts might be correlated with earthquakes and/or volcanic activity.
The atmosphere, being a fluid, is affected by the moon, resulting in an Atmospheric Tide, a wave that propagates through the atmosphere. However, the increase in atmospheric pressure that occurs at the front edge of the wave is so slight that it is hard to detect from the myriad of other waves that are always present in the atmosphere.
The moon has no statistical effect on the weather but there is a minute effect on atmospheric pressure.
"Tides are raised in the earth's solid crust and atmosphere as well as in the oceans. Every body in the universe has some tidal effect, however small, on every other body. This effect is directly proportional to the mass of the body causing the tide but inversely proportional to the cube of the distance between the bodies. The earth's nearby moon is about 2.17 times as effective as the more massive sun in raising tides on the earth, even though the sun exerts a much greater total force on the earth than does the moon. Thus, the moon's proximity explains its dominant role in creating tides." See this link for more details...
http://www.infoplease.com/ce6/sci/A0861550.html
There are atmospheric "tides" but they are mainly due to heating of the upper atmosphere by the sun. They cause a diurnal pressure variation, with the maxima around 10 a.m. and 10 p.m. each day, with minima around 4 a.m. and 4 p.m.
Here is a link explaining this phenomena...
http://www-das.uwyo.edu/~geerts/cwx/notes/chap01/diurnal.html
And here is an interesting fact...atmospheric tides were a factor in the period of rotation of the planet Venus, which has a much denser atmosphere than the Earth. Here is a link to further explain that...
http://www.imcce.fr/Equipes/ASD/Venus/venus1_en.html
However you raise a very good point as I doubt any of these forces are input into the now infamous computer global warming models!