In the hypothetical example of the OP, the 1099-DIV shows a non-qualified dividend.
I don't think this would happen; you would receive a 1099-DIV that shows a qualified dividend.
In other words, dividends that could be qualified are presumed qualified.
It is up to the taxpayer to disqualify his own dividends on the tax return.
Note: never experienced this situation personally (that I can recall), so I can't say with absolute 100% certainty.
