Just have it an unofficial faction organized only by the Hawkbat tag. Have all the members still have the ID of the people they've just defected from - and have PVP by permission only (ask the guys you're going to attack before attacking). Naturally, since rep to the faction of origin may be hostile, you won't be able to land on any of their bases, but there are tons of freeports and semi-lawful bases to dock at. The only issue would be ZoI but so long as you've got RP justification, everything should fly fine.
