Why don't we just write it up like this?
Rend Fallen (Ex): An elephant bird automatically deals an additional 1d6 points of damage to a foe it knocks prone with its overrun attack. If an opponent goes prone for any reason (perhaps as the subject of another elephant bird's overrun attack) in an area the elephant bird threatens, the elephant bird can also rend the victim as a free action (treat as the elephant bird's attack of opportunity for the round), even though it had nothing to do with overrunning the foe.
It shouldn't do a ton of extra damage, as it only threatens adjacent squares, and it lacks Combat Reflexes.