For now, a moon mission runs on the time of the country that is operating the spacecraft. European space officials said an internationally accepted lunar time zone would make it easier for everyone, especially as more countries and even private companies aim for the moon and NASA gets set to send astronauts there.
That’s true, but it would still result in a shift in measured time between the moon and earth.
Yes. But to make up that difference, clocks on the moon would have to tick slower than clocks on earth to keep syncronized. Or they could do what you suggested, and add a leap second in every once and a while.
Either leap seconds or relativistic timezones. I don’t know which is worse.