Sadly, you generally have to make it over the fence and close enough to an actual White House door that you can't be tackled before they'd shoot. And the WH is so far back from the fence that there's nothing she could throw over the fence that would pose a threat to anyone. It would just land on a patch of grass.
Oh yes, the security is so good there, that if you merely try to toss something...even explosives...over the fence, you can't actually harm the President or the WH itself, so you can get away with a whole lot - and the looney haters KNOW it, too.