Is "The Valiant Woman" disrespectful to the Catholic Church and the priesthood? Give reasons.

It is possible to respect an institution and at the same time criticize components of it or the people involved in it. In "The Valiant Woman," J. B. Powers presents two individual priests who have different attitudes toward their calling and toward the people who make it possible for them to pursue it.
The visiting priest, Father Nulty, is a self-centered man who promotes this behavior in others; he also seems to be a misogynist. He finds the story he tells about another priest who told lies in order to trick his housekeeper into leaving her position amusing; one of those lies was pretending to have illegitimate children. Apparently, Father Nulty does not think about the consequences of the woman losing her job.
Father Firman seems to appreciate Mrs. Stoner but also fears her. He is portrayed as a weak and passive person. While he fantasizes about the idea of having a docile housekeeper (using a racist stereotype of Asian women), it seems unlikely that he will use false pretenses to unjustly dismiss a widowed woman who needs a job.
Both men—and perhaps, by extension, the Church—accept the idea that a priest must have a housekeeper and thereby promote the kind of arrangement in which Father Firman feels trapped.