According to a deleted scene, the shuttle was named Copernicus.  The registry number of the shuttle was somewhat illegible on screen, but appeared to be BA-3(?). The roof of the craft included the identification numbers DF723 on the outside and FAB.1X456.31 on the inside, though it is unlikely that either of these was the registry of the shuttle based on their locations.
The Enterprise's main shuttlebay was three shuttle storage decks high, with Takayama-type shuttles stored on the 3rd level on the port side. There were at least four shuttles stored there. Military shuttles were stored on the 2nd level.
For the sequence showing shuttles tumbling from their port side perches, shuttles labeled 7 were seen twice, once in the foreground and once in the background (as shuttle 18 here). Shuttle 4, situated between the two and nearer to 16 while the shuttle labeled as 19 here was closer to shuttle 7, was unlabeled with a name and might have been a CGI creation.