While bringing in goods was something of an issue for DS9, we never heard of risks or difficulties associated with delayed fuel shipments. It appeared that the onboard power generators were quite capable of operating systems such as personnel transporters (for apparent daily commuting of large numbers of Bajoran laborers, back in ye olden days), shields and phasers (for fighting multiple starships), even in a siege situation...
Where the comparable powerplant of K-7 might be located is an interesting question, with many possibilities. K-7 is significantly smaller than DS9 regardless of what exact size we choose for the two ill-scaled structures, within the apparent limits. It's not smaller than Kirk's ship, though, and could plausibly mount a "starship-category" powerplant or three in, say, the hangar section, with significant fuel tankage along the booms.
The station manager's office can only receive two visitors at a time beaming in, already three visitors will be a 'problem'.
Which is a very good limitation to have for a station manager's office! The mall transporter might be quite capable of receiving parties of 58 people at one stroke, though.
This is assuming that the platforms on the station handle the transport, of course. Kirk could have beamed in with at least five friends if he didn't bother with Lurry' platforms - and with twenty-one if the larger units are assumed to exist. A problem, definitely, but only for Lurry...