Actually, the full moon is illuminated by our same Sun, so the same Sunny 16 rule is of course applicable. However, the overall albedo of the
Moon is around 0.12, so the true surface is actually darker than we like to see it in pictures. So opening one stop does please us more, looks like we think we see it, bright in black sky.
But that one stop is only for a full moon. When the moon is less than full (side lighted at quarter, or back lighted as crescent - instead of frontal lighted at full), we need to open more yet (the angled light does not reflect back to us as directly or efficiently). So yet another stop open for gibbous (2 stops now), and yet another stop for a quarter (which is 3 stops from Sunny 16), and yet another stop for a crescent (or maybe two more stops for a thin crescent). We gotta do what we gotta do.
These are just starting points, of course make it look like you want it to look (you have plenty of time). Spot metering with a long lens sometimes can work (maybe open a stop), but the manual exposure can be controlled better and easier. Metering of mostly black sky will be overwhelmingly overexposed.