The ship would keep local time because that is easy to measure and fits with what people are doing. The point of establishing Moon time is that it is hard to ke

The ship would keep local time because that is easy to measure and fits with what people are doing. The point of establishing Moon time is that it is hard to keep using non-local Earth time. Keep in mind that there is difference between clock measuring seconds, and converting to days and times that requires choosing a calendar. The ship could use origin calendar, destination calendar to make people familiar, or some universal calendar. The problem with destination is that it won't sync properly until arrival. Or maybe they skew the calendar on the last leg. Also, don't need high relative velocity to have a difference. Considering relativistic space travel may be impossible; the only may be proposed warp drives.