[Alneyan]

A Battlecruiser with 20 Minesweepers will have exactly 0kt for other components (a mere 600kt is hardly enough for twenty 30kt components), so it isn't going to be built anytime soon, let alone move anywhere.

A Battleship with Emergency Propulsion V and Propulsion Experts should get the job done, though, and will sweep several hundred more mines a turn. Of course, most people don't lay a full 100 mines on every single sector.