Why do you think Melville chose the unnamed Wall Street lawyer as his narrator?
The unnamed narrator is in many ways the opposite of Bartleby: an optimistic and stable man, the lawyer seems to be a completely devoted contributor to both society and the economy in general. Thus, it's possible Melville chose the lawyer as his narrator to more dramatically emphasize Bartleby's stagnation and listlessness. Unlike his employer, Bartleby seems completely apathetic about his work, and seems to have no faith in the occupation that he has chosen for himself. Indeed, Bartleby seems to be simply going through the motions in life and is uninterested in his work. As such, the lawyer's optimistic work ethic and contribution to the work force is, in some sense, deconstructed. Therefore, we can hypothesize that Melville chose a stable and normal lawyer as his narrator in order to call into question, through Bartleby's apathy, the robust American work ethic represented by this lawyer.
This answer is perfectly legitimate, but ignores the deeper significance of Bartleby’s “I prefer not to.” Melville is trying to dramatize the difference between goal-driven action and personal choice. While “existentialism” is not exactly the right word to describe this difference, Melville’s entire canon serves to show the consequences of choice—choosing to go after Moby Dick, or Bill Budd’s choosing to “bless Captain Vere” rather than cursing his fate, etc. The unnamed lawyer in Bartleby the Scrivener has chosen to conform to the written and unwritten behavior of the business world, and as such represents all the nameless persons whose choice-making is driven not by personal conscience but by almost mindless conformity to the mundane. Melville himself is a conscious “chooser” rather than a conforming author.