rsuggit,
Your answer is pretty much the same as mine I gave earlier. Don't get distracted by the moisture content, it does matter if the air is very dry or very moist, as long as it's not saturated, idealized air parcels will always warm or cool at the DALR (approx 9.8C/km).
Remember that idealized parcels will retain their moisture content and dewpoint temperature until the air parcel temperature reduces to the dewpoint. As the temperature decreases condensation occurs and latent heat is released due to water changing phase (vapour to liquid) which modifies the lapse rate.
Also I think there might be confusion regarding the difference between the inversion resulting from the presence of a synoptic high pressure and the one associated with radiative cooling. Both have the same physical effect, in that they stop buoyant plumes and particulates leaving lowest layers, but it is not true that surface cooling is required for an inversion to form - note that they can persist during the daytime in summer. Similarly you have nocturnal inversion form under other synoptic conditions, all you need is a clear sky and light winds overnight.
However, the fact of the matter is that you do tend to get both the anticylonic inversion and the nocturnal inversion occuring together since anticyclonic conditions do favour clear skies and light winds, which doesn't help the understanding of things! If you have access to tephigrams or skew-t diagrams it is sometimes possible to see both synoptic (anticyclonic induced) and nocturnal inversions separately as the nocturnal one tends to exist near 900hPa and the synoptic one is around 850-700hPa depending on the conditions. If you have a time evolution you'll also see that the nocturnal one tends to be eaten up during the day.
Also as one of my collegues in the Atmos. Chem group pointed out, there is also an increase in photochemical smog during the day that drives down the viz over the daylight hours. Some talk of OH and ozone sulphates and aqueous phase blah blah. Chemistry isn't really my thing!
Anywho, I hope all this 'dry' meteorology is helping.....
Cheers,
Gareth.