I am not seeing a problem that we have not already encountered and solved in similar circumstances.
We already deal with different clock speeds between Earth, low Earth orbit, Mars landers, and even the various probes we have sent to the gas giant planetes and beyond, even out of the solar system into interstellar space.
We already have bases in Antarctica that have periods of light and dark that are far different than the circadian rhythm that human bodies are tuned to.
We already have astronauts working in low Earth orbit for up to a year at a time, seeing the Earth below going from light to dark and back again roughly every 90 minutes, while they are in light a bit more than darkness.
All of this is done with standardization "Universal Time", and we are pretty darn good at it, by now.
So, while the Moon probably needs some sort of system for relating its various cyclic phenomena phases to Universal Time, I don't see any value to having all of that somehow predicated on something like just one of those cycles, such as when the Sun angle at a certain location is some number or the Moon's location crosses the Earth's meridian for Paris. Yes, those numbers may have some importance for some functions, but having the astronauts wearing Rolexes that run on "Moon time" seems ridiculous.