There is literally zero need to have long-term tracking of "Moon time". For all long-term (day/week/year scale) you can work in whatever Earth time zone suits y

There is literally zero need to have long-term tracking of "Moon time". For all long-term (day/week/year scale) you can work in whatever Earth time zone suits you best, and when the clocks will drift - no matter if that drift is due to relativity or plain old inaccuracy - just sync them to Earth time. Yes, an isolated non-adjusted clock on the moon would have slowly accumulated some drift from the correct time, but who cares about that? For use cases which need extreme accuracy, do explicit adjustments for the time dilation in these specific conditions, just as we do for GPS satellites, instead of trying to define a single moon-wide lunar second or some offset from UTC.