I wonder if Lels used the name Nightingale even when she was still a bard, pre-Origins. Highly doubt it though, since she never mentions it.
I'm sure her code name connected to her job as The Left Hand (seeing as how her full name is Sister Nightingale) rather than a bardic title, but then again one might wonder why she didn't choose something that didn't symbolize poetry and singing since her job as Left Hand only involved spying and killing.
She was the Divine's personal assassin as Left Hand, no seducing, no singing or sweet-talking the target, just cold blooded killing. Ain't nobody got time for that. Hence why she's against throwing a party during a war table mission if the Quizzy chooses the Champoin spec. Or refusing to join Josie and Cullen for tea. Or anything other than her work as Spymaster.