"Trials..." indicates that most of the station is unmanned storage compartments; both episodes show what appears to be a civilian in command of the station, and no Starfleet personnel appear to be present apart from those from the Enterprise and Defiant. This indicates to me that K-7 may fall into the same sort of category as DS9, i.e. a station not owned specifically by Starfleet but operating under its control or authorisation.
Some have suggested that the "K" of K-7 indicates that the station is one of at least seven situated near Klingon space, and that there would be a R-1 near Romulan space, T-1 near the Tholians, etc. This is quite a nice theory, although we have seen the stations near the Romulan neutral zone named on a map and they were designated "Outpost" rather than "R". Then again, perhaps the first-letter designation only applies to civilian facilities?
Clearly, there is a lot as yet unknown about this station. Alas, it's not likely that we will get to see K-7 or its ilk again.
Most of the figures for K-7 are invented by me, and are intended to
be in line with civilian ownership or control.