Though we often think of the protagonist as the lead character in a story, a protagonist can also be an important character or a character, according to the Merriam-Webster dictionary, who is the "supporter of a cause."
Given this definition, the narrator is definitely a protagonist in the story. He is not only telling Bartleby's story but has an important role in it. A lawyer, he hires Bartleby to do copying, and for a long time after Bartleby decides he "prefers not to" do anymore work, tolerates his presence in the office. Even when the lawyer feels forced to move offices and Bartleby ends up in the Tombs, the city jail, the lawyer visits him and tries to ensure he is well taken care of.
Bartleby takes up a disproportionate amount of the lawyer's attention, and the lawyer feels a sympathy for him an average employer would not for an employee who stopped doing any work. Because the lawyer is telling the story, we also feel a good deal for sympathy for and curiosity about Bartleby. The lawyer has the sensitivity to perceive that Bartleby is a troubled soul. Rather than write Bartleby off as crazy, the lawyer becomes a "supporter of [his] ... cause."
The lawyer even continues to be curious about and investigate Bartleby after his death, discovering his job at the Dead Letter Office in Washington. He fashions Bartleby for us as a symbol of the wider human inability to communicate effectively.