How would you? Periodic leap seconds, or redefine the second (which would have impacts on many other units of measurements derived from it – are you prepared to also introduce lunar meters, for example)?
The compensation for the relativistic effects just makes sure that the second on the Moon is the same as the second on Earth. According to the article another commenter linked (https://www.ipses.com/eng/in-depth-analysis/standard-of-time...) this is already routinely done on GPS satellites:
> the only corrections made on atomic clocks located on satellites are very small adjustments to ensure that they remain perfectly synchronized with atomic clocks installed on the Earth (usually to correct drifts due to relativistic effects).
this takes care of the effect mentioned in the article. However the more relevant (and debatable) question is if "Moon time" should also observe the leap seconds (which are introduced to account for variations in Earth's rotation, so have nothing to do with the Moon).
This only works on the GPS satellites because they provide the GPS time for Earth's benefit, not their own.
For all internal operations requiring high precision, they'd actually have to keep their own time reference (or calculate a dynamic offset from "GPS time") or they'd get unexpected results since the atomic clocks they carry run fast with regards to the SI definition of a second.
(It could be the other way around, i.e. the clocks running correctly with regards to the satellite's frame of reference and an offset being applied to the signal, but I suspect skewing clocks to fit the Earth-based frame is easier.)
