You're not going to be able to. Think about it, that if you have no way to verify that you are the account owner, what a can of worms it would open if any rando could come along and get someone else's twitter account shut down.
Why do you need the account shut down? If it does not include personally identifiable info, I don't see the need. If it does contain personally identifiable info, then that might be your gateway to help twitter staff identify you as the account holder.