The NCC-K7 was not referenced in dialogue, nor was it seen in "The Trouble with Tribbles". Rather, a miniature of the vessel was created for "Trials and Tribble-ations." Footage of the shuttle could be seen in the DS9 Season 5 DVD's special features that look at the making of the episode.
It is unclear if the registry was something assigned to the shuttle only or the registry of the space station. Star Trek Star Fleet Technical Manual depicts a shuttle assigned to Starbase 1 with an NCC registry, NCC-6038⁄5. The shuttles not assigned to starships that appeared later on in the remastered episodes of Star Trek, such as the Picasso, were registered differently as "SB11-1201⁄1". It should be noted though that the starbase the Picasso was assigned to was not a space station.