To clarify, there are two independent issues here:
1) Disciplining clocks to mark the passage of time on Moon the same way they do on Earth in a synchronized fashion -- effectively ignoring / compensating for relativistic gravitational effects.
2) Choosing a particular Earth timezone that (all) people on the Moon will use.
To me, doing anything other than 1) is just plain crazy. I don't see any good reason for not keeping in-sync with the passage of time on Earth. The relative relativistic effects are tiny and completely unnoticeable to humans over the course of a day. The only exception is scientific measurements that need a highly accurate and precise measure of the passage of local time, which should use local atomic clocks.
On 2), who really cares? People should just specify the TZ they are using if there is any confusion. I imagine a Chinese mission would likely use Beijing TZ, whereas an American mission would use Eastern, and so on, as their sleep schedules should likely tie to their support people on Earth. There is no strong reason to try to force any one particular choice for everyone on Moon.