Hmmm...
Thinking a little more, Mr. Lucky will have some coding to do. Blocks are indexed based on their hash, so when he generates that second all-zero hash he's going to have trouble with the current implementation. Actually, he'll have trouble before then, because if the target is low enough there won't be enough unique hashes...
(and before somebody asks: YES, there is a very small chance that two blocks will be found with the same hash. And NO, that is NOT a problem that needs to be solved, it is so improbable it is not worth worrying about).