Black holes emit jets (through an as yet poorly understood mechanism) as seen in active galactic nuclei, e.g. Hercules A. From this, it seems straightforward, even trivial, to build a Death Star: point a black hole in the right direction (moving it first if necessary) (building it first if necessary), then dump a bunch of stuff into it (assuming that's the mechanism that causes jets).
How large a black hole is needed? Does it need to be a supermassive black hole or do stellar mass ones also produce useful jets (microquasars)? How powerful / deadly is the jet -- how close does one need to move the black hole? How much stuff needs to be dumped in to power the jet? I suspect black holes are very efficient at E=mc^2.
Some galaxies might be so unlucky as to not need astroengineering. Their SMB jets might have somehow (perhaps after merger) gotten naturally aligned with the disc of the galaxy (the black hole is "on its side", like Uranus), so as stars orbit the central black hole, its jets irradiate every star system in the disc. Tell a story of a civilization which sees a jet approaching.
How do you move and point a (possibly supermassive) black hole? If only stellar mass black holes are needed, then it might be easier just to find one that is already pointing the right direction.
Moving and pointing black holes seems so difficult that it's almost certainly easier just to move the target, perhaps into its sun. (Though maybe the target has defenses against that, so a ranged weapon is desirable.) However, sterilizing an entire galaxy as described above does seem an appropriate use case (assuming it even works: life often finds a way).
A K3 civilization might also use jets not just for warfare but also as a power source.
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