Converting Mimas would be good... But how do you STEER an actual moon? That's a good point... And I think they needed all that room inside Death Star for the power subsystems... Think about it.... The power generation needed to power a DEATH beam need a killer refrigeration system (pun intended), that would be really huge.
On a side note... We could save ourselves the problem of steering Mimas for targeting if we could generate the beam on the moon core, and simply redirect it thru the apropriate tunnel (the moon would have tunnels going from the core to many directions). There you go, no fuel costly moon steering necesary anymore.