How does the narrator of "Bartleby, the Scrivener" perceive himself? Is his assessment of himself accurate? Why or why not?

Bartleby's employer, who narrates the story, seems to view himself as a man of both common sense and compassion. His perception is arguably justified, though perhaps, given the tragic fate of Bartleby, our positive assessment of the narrator should be qualified to a degree.
Anyone who has read the story probably regards its scenario as one of the more bizarre in literature. A man who slowly and systematically refuses to cooperate with the people surrounding him, in spite of being treated with patience and kindness by them, and who doesn't behave in a violent nor obviously psychotic way, is an anomaly. And yet, the realism with which Melville invests the story carries one along and gives the narrative the feel of truth, not just in the details, but in its overall message and as a metaphor for the "human condition," trite as that might sound.
Normally of course, at the time "Bartleby" takes place, and in today's world, an employee who persistently (or even once) refuses to carry out tasks assigned by the boss would be fired in short order. The lawyer refuses to do so because he's genuinely compassionate and is disarmed by Bartleby's innocence and total lack of guile. The details by which the lawyer explains his actions—or lack of action, more precisely—strike the reader, for the most part, as being reasonable under the circumstances. But in spite of the lawyer's patience and sympathy, one feels that he has not really attempted to get to the bottom of Bartleby's problem—although, perhaps, this only appears to be so from our 21st-century perspective.
Though psychiatry didn't yet exist as a special branch of medicine and "Bartleby" was written several decades before psychoanalysis began with Freud, the lawyer, even in the 1850s, could and should still have called in a physician to examine Bartleby and perhaps give some clue to his behavior, as well as give a diagnosis of sorts. However, if this had happened, an order probably would have been given for Bartleby to be taken to an asylum. At the time, such places kept "lunatics" under conditions that were barbaric, by our standards. So this option probably would not be any more humane, or would be less so, than just allowing Bartleby to deteriorate as he does in the office and then finally taken to prison where he starves himself to death.
To criticize the lawyer for not doing more than he does, however, is probably to miss the point of the story. The final line "Ah, Bartleby! Ah, humanity!" makes it clear that Melville intends the story as a parable, as a metaphor of man's alienation. Bartleby withdraws from life because he is, apparently, confronting an existential question about the "purpose" and "meaning" of any sort of activity, or of life itself. He may be a man seized by the same "hypo"—a nineteenth-century term for depression, a sense of emptiness or incompleteness—such as that which Ishmael describes at the start of Moby Dick, and for which his "cure" is to go to sea. Bartleby's situation is similar, as well, to that of Parson Hooper in Melville's close friend Hawthorne's "The Minister's Black Veil." Hooper withdraws from life by placing a veil in front of his face, cutting off, in a radical way, his connection with other people. Hooper's intent is presumably to teach others, to tell them that every man and woman is wearing a kind of veil without an awareness of it. But as with Bartleby, Hooper is expressing his own alienation from the community and from life as a whole.
The narrator of "Bartleby," though on the surface baffled and shaken by the whole story he tells, still comes to an understanding of the deeper human problem represented in the behavior of his hapless scrivener. And he does, therefore, accurately see himself as a man of compassion and understanding, despite the tragic outcome which he is unable to prevent.