The definition of a second is as follows [0]:

  The second is defined by taking the fixed numerical value of the caesium frequency ∆ν, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s−1

But with the Moon's lower gravity, time flows very slightly faster (a moon-second is 0.99999999999848 of an earth second according to a sibling post). I'm no physicist, but my assumption is that the SI definition of a second would still be true on the moon (at least relative to the moon's reference frame). So it would make sense to me to leave a second defined as a SI second, regardless of your reference frame because that's how humans, science & technology measure time passing.

I think the bigger calendar problem with space travel is the differences in the length of a day (or sol), as we know it's 24hrs on earth, Mars is 24h 37m (not sure about the moon since it's tidally locked its ~27x24hrs). The drift of days between Mars and Earth is going to be far more noticeable than a tiny fraction of a second due to relativity.

[0] https://www.npl.co.uk/si-units/second

That might be convenient for the purpose of timekeeping, but then you would also have to redefine all sorts of physical constants, which could be problematic. Oops, our lander crashed because we were doing calculations using Moon seconds, but the gravitational constant G was in Earth seconds. Okay, the difference probably isn't enough to matter for that case, but still any physics calculations where that precision is needed (say interferometry?) you'd have to keep track of which measurement system you are using. In some ways it would be worse than US -> SI because they would be close enough that it wouldn't be immediately obvious you made a mistake.

I'd much rather have SI be universal than our calendar.

That’s great until we occupy a smaller moon and need to reduce the length of a second for that moon’s time dilation.