In 3025, Jumpships go to empty Interstellar Mediums, if they absolutely know they will be able to make a second jump to safety. Since most Jumpships are held together with duct tape and crossed fingers, most Jumpships won't do that.
Then you have a few Jumpships (some House vessels, a few major companies) that are in good enough shape to make those jumps, and fewer of those do so repeatedly (and charge a nice fee for the capability).
And then there's Comstar. Their Jumpships are in good shape, but there's only a few of them. So if you need reliable shipping capacity to your hidden base, they are the ones to talk (and pay lots of money) to.
As to recharging, engines can have small spikes in their power output that isn't an issue with weapons fire, but the Jump Core will complain about. By using Batteries you can have a smooth power output to avoid the KF Core barfing itself all over. What likely happens is the base uses a small fusion power plant to recharge the batteries, and then the batteries to charge any nearby KF vessels. Also by using recharge batteries, you get a discount to the time needed to recharge (IIRC +2 on the success roll, offsetting a -2 on the roll for using 150 hrs instead of 175). This could be due to recharging producing heat in the KF core, and the direct connection allows using the station's heat sinks for coolant, instead of just the Jumpship's.
A comparison would be the Liquid Mirror telescopes in use today. They use a reflective liquid (mercury, gallium, or silver in a glycol solution) in a flat pan, and the pan is spun. The spinning liquid forms a parabola shape. However small power surges from the power lines will distort the spin rate enough to make the images blurry. By charging batteries and running the spin motor off the batteries this small fluctuation is removed.
KF cores are poking holes in the universe to get to the other side. You don't want to make a
mistake.