The question is not, "What does Bartleby want?" It is never clear that he wants anything at all. The problem all started here:
In this very attitude did I sit when I called to him, rapidly stating what it was I wanted him to do — namely, to examine a small paper with me. Imagine my surprise, nay, my consternation, when, without moving from his privacy, Bartleby, in a singularly mild, firm voice, replied, “I would prefer not to.”
So it's not what Bartleby wants; it's all about what he doesn't want.
There is no question that something is seriously amiss with Bartleby the scivener. But, in this very Kafkaesque tale of the world of business, is there something wrong with Bartleby or is there something wrong with the whole world Bartleby happens to inhabit?
Consider his life. He writes and writes and writes. And what he writes is all copied as if by a machine. Nothing original comes from Bartleby. Nothing. The story takes place before carbon copies, Xerox machines, or computers, where forms had to be hand written and reread over and over and over. And Bartleby plainly states, “I would prefer not to” again and again and again. He shuts down and just refuses.
What does Bartleby want? He wants not to.
He is the reverse image of people who go to work every day at jobs they don't care about doing repetitious, meaningless things and never question what they are doing with their lives for a second.