I've been reading Christopher Hibbert's biography, Wellington: A Personal History, and there is an interesting tidbit in there.
When Wellington captured Madrid from the French in 1812, he was showered with various gestures of gratitude (including vast sums of money), and the idea came up to add an augmentation to his arms. Hibbert states that the Duke's brother, Richard Wellesley, 1st Marquess Wellesley, proposed that the French napoleonic eagle be added.
This would be in line with the more famous and far older Howard augmentation -- i.e. adding an augmentation referencing the vanquished foe, in Wellington's case the French Empire.
However, Wellington apparently thought this would be a bit over the top and pretentious, and so opted for the union flag instead.