OK, I have the answer. I was successfully aligning the text within the cell. However, it appeared that I was not because the CaptionPanel was left-justifying my FlexTable.
Note: I figured this out by assigning a high contrast background color to the FlexTable cell.
My solution (perhaps just a workaround) was to use setWidth to make the FlexTable cell fill the width of the CaptionPanel.
Thank you, Sam, for the conversation. Your feedback helped me sort out the issue by helping build confidence that what I was doing with the FlexTable was correct.