I don't see what's wrong (that is, incorrect, ethics aside) with assuming that Mudd murdered people only when knowing it wouldn't matter. It would be solely up to Mudd to decide whether to make a timeline stay or be erased. He erased all those where he killed Lorca. He eventually erased even the one where he won but killed Tyler. So Mudd's character hinges on that "eventually": would he have reset if not for Burnham's clever suicide?
Knowing how to hack into the Discovery might count in his advantage: perhaps he was correct in stating the ship was embarrassingly poorly defended? For all we know, those faults were now corrected, solely thanks to him.
Ultimately, though, it's up to Lorca to decide whether Mudd will fly or fry. Kirk, too, had that power: he let murderers go, and at least once condemned Mudd to life imprisonment (that is, solitary confinement until death) outside the UFP judicial system (which, let's again remember, does not believe in punishment of any sort from TOS onwards). Lorca let Mudd walk, and Kirk let Khan walk. This may come to bite them back later, or then not. Either way, it's par for the course for how justice and starship captains work in 23rd century Trek.
Timo Saloniemi