The game would be almost impossible to playthrough LS then. And also, most of the people you kill are your enemies, who need to be brought to justice. Hence, a Jedi.
You may be the judge and jury, but executioner as well? That seems a bit too evil. In K1 during your initiation for Jedi there's a similar judge/jury role, but the perpetrator is returned to the enclave. Combined with @metadigitals idea a la "Bloodlines" (exp for quests only, even though there is plenty of combat to be had if you so desirre) and my idea about more enemies yielding, the LS jedi could be about returning those offenders to court for trial...but in any case: killing enemies at an alarming rate, that's just too much bloodshed for a LS I think.
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It IS evil. Even in the frontierland of the Wild West of the USA post Civil War, it was encumbent upon the Marshals to bring evil-doers back to justice under a judge (viz. the eminent historical exegesis Hang 'Em High").
Taking this idea further, non-lethal combat could actually become a feature: deciding to disarm an opponent rather than to kill would be a (reversible) decision taken prior to combat: difficulty would necessarily be increased (it's a lot easier to lop someone's head off than to knock them out); success would mean that the bad guy(s) are incapacitated; choosing to change from non-lethal to normal combat would have no deleterious affect on the battle, but it would mean DS points when the opponent(s) were killed.