I literally just watched this comparison video the other day between watching videos on casemaking.. I'd have liked to have seen a comparison done on both sets of jaws as internal measurement is often less nice than external. The cheap measuring tools are good enough for their purpose.
Perhaps it did migrate, but someone initially stated that 4-jaws would be better, without referencing hobby machining.
I think there is a reason why self-centring 3-jaw chucks are the usual choice for watchmaking. I rarely do any eccentric turning, but I do have a faceplate which I can use.
Back to the original post: I can recommend getting a Burnerd or similar quality 3-jaw (German, Swiss, English, American). The Sherline chuck which jdrichard describes sounds promising too.
Further to John's remarks,
-I would advance the hands by 12hrs, if the fault is in the calender mech then it should loose time during the day.
-Note how long it runs on bench following a full day on wrist.
This points to the area the fault is at.