There is at times disagreement over what characters fit the definitions of flat and round characters. Understanding the definitions is helpful.
Aside from flat stock characters (the foil) and stereotypical characters (mean stepmother), flat characters who are poorly or briefly developed have rigid physical descriptions (with green eyes that flashed golden (always?)); have a narrow range of emotions, feelings, thoughts; have simple motives; lack psychological (the interplay of cognition and emotion) depth and complexity. In short, flat characters are developed within narrow and restricted patterns and, in addition, may be identified by a particular single quality, idea, or trait (shifty eyes).
Round characters, on the other hand, are developed with, envisioned with, and endowed with a flexible physical description; a full range of emotions, thoughts, feelings, ideas, motives; they have complex psychologies with depth and complexity. In addition, they are not identifiable by one particular single quality, idea, or trait.
Captain Brown may threaten to be a flat character at his first introduction since his main function then is to declaim his poverty in the streets. However as his involvement in the ladies' lives and in the story deepens, he is shown to have a full range of characteristics, as enumerated above. An example is the range of emotions and reactions he experiences around the combative readings of Mr. Boz and Dr. Johnson.
Further, a deep psychological struggle is introduced as he distracts his attentions from his friends at church until after he has attended to his elder daughter. Therefore, Captain Brown is a round character, but a subsequent discussion may address the author's skill in portraying her characters, even though they are round.