In particular, does the pull of the moon affect the amount of water vapour in the Earth's atmosphere? In other words, if the moon were larger, or orbiting more closely to the earth, would the increased gravity cause the composition of the atmosphere to change?
The atmosphere is continuously mixed by turbulence, so no sorting of its ordinary components can occur easily and rapidly. Twice a day change, anyway, is far too fast for anything but some tidal effect on pressure and thus on temperature. Ok, these might indirectly slightly modify some thresholds and balance e.g. between states change (water) and some atmospheric chemistry, by a fraction of what day/night change does. I wouldn't call that "change of composition", though.
The long term influences upon the Earth's atmosphere are additions, subtractions and internal variations. The additions consist of gravitational attraction of passing gas molecules from space, which is insignificant, and outgassing from the solid Earth. The latter is 'new' helium from radioactive processes, and volcanic gasses which are mostly long-term recycled from the surface via subduction zones.
Subtractions involve loss of gass molecules by thermal heating in the very tenuous outer regions of the atmosphere. Some of the lighter gasses, mainly hydrogen and helium, attain energies which allow them to exceed the escape velocities. This is why our atmosphere hasn't built up much higher concentrations of the light elements over time.
Internal variations are such geochemical and biological feedback processes as greenhouse gases influencing the temperature, and hence evaporation / ice accumulation, etc.
None of the above have anything to do with pressure fluctuations caused by the moons orbit.