I was thinking of using the coilgun but in reverse, charging the capacitors (or flywheel as you suggested which is also a good way to store energy) and decelerating the spacecraft.
How did you get 50 ms? I'm getting 200 ms:
acceleration = velocity / time = 4000 m/s / 0.2 s = 20'000 m/s^2
distance = acceleration * time^2 / 2 = 20'000 m/s^2 * (0.2 s)^2 / 2 = 400 m
Maybe we don't need to accelerate the spacecraft to 3.2 km/s in one go if we do it in small increments and with a longer coilgun.
If we can split the acceleration in 3 and use a 1 km long coilgun, then the time could be increased to 2 seconds and reduce the acceleration from 2000 g, to only 50 g. Maybe a well trained astronaut could take 50 g for 2 seconds? Maybe it's possible to build an apparatus to keep the eyeballs in place and prevent them from escaping the eye sockets.
Good idea. Maybe if we keep the slingshots happening at a constant rate, the orbit of the station will be circularized every 27 days (the orbital period of the moon).
EDIT: if anyone is interested in seeing how the "double lunar gravity assist corn" was made, here is a youtube video I just made of the simulation using Bugale N-Body Simulator:
The mass of the earth, the mass of the moon, the velocity of the moon, the distance of the moon from earth, the diameter of earth and the diameter of the moon are all simulated. In other words, that orbit is actually possible in real life.
Btw, the yellow trail (behind the spacecraft) has a width of 400 km.
EDIT 2: I think that, once we have a coilgun in space, all space debris becomes valuable because it can be deorbited and most of the orbital energy added to the station.