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.
The space shuttle had a four-second window to start its entry burn to reach the runway back on Earth. I don't think the discrepancy in a second on Earth versus the Moon is going to mean the difference between a crash or a success. No space-faring vessel had any tolerance as low as .0000001
I'd much rather have SI be universal than our calendar.