My rule of thumb is, if you can solve a problem using conservation of momentum (linear or angular), do that. If you can’t use momentum but can use energy, do that. Only if you can’t use either do you start breaking out other methods.
If energy and momentum would either one work, momentum wins out, because it has fewer “gotchas”, or opportunities to make mistakes. It can’t hide, and there are fewer opportunities to screw up the reference frames.
Of course, this does require recognizing when you can use conservation of energy and/or momentum.
That rule of thumb pretty much corresponds to my mental approach, though I don’t keep an explicit list in mind.
I tried solving the problem using CoE, but there’s a tricky part that I don’t have a way around.
The energy expended by the gun should be equal to the final KE in the starting frame of the ship (since the initial KE in the starting frame is zero). The gun energy is easy to calculate: it’s just KE_{gun} = 1000000 \cdot \frac{1}{2} \cdot 0.145 \cdot 1000^2 = 72.5 GJ
The KE of the ship is also easy, given the rocket equation. It’s just KE_{ship} = \frac{1}{2} \cdot 12000 \cdot 2571.3^2 = 39.67 GJ .
The KE of the balls is trickier. They range in velocities between -1000 and 1571.3 m/s. You can integrate across these, but you need the mass density to do so. And it’s not constant, because the balls are not evenly spaced. They get farther apart (in velocity space) the faster the ship got. Maybe there’s an obvious method but it’s not coming to mind.
I think if you’re doing it with energy, you just deal with one ball at a time, find the \Delta v for the ship from that one ball, and then add up all the \Delta v (either one at a time manually or in a spreadsheet, or via some sort of summation formula).
Well, that’s just cheating 
. There are enough balls to treat the problem as continuous, but the mass density is a problem.
In any case, there are some interesting effects at work. For the later balls, the remaining KE is greater than what the gun imparted. That’s because previous balls gave some of their KE to the ones still on the ship. The balls that ended up at low velocities (like the ones fired when the ship was at 1000 m/s have zero final KE) contributed the most. It all adds up to the right value, though.