I think you could show that on the "open" market these AAdvantage miles are sold for less. See the listing at EBay - item #2290639693. $499.99 for a 25,000 mile award. Not a big savings over what AA said it was worth, but nonetheless it shows that you could find a lower price for the same miles.

I once won a 100 disc CD changer. The company awarding the prize had it valued at $1,000. As it happened the next model had already come out and the value of the CD player was less than half that. Easy to show the IRS the actual fair market price by enclosing an ad with my tax return.