Yes, this is possible to do via script and get Balance Due data from API2.
But i think, this is very risky, we must make sure the balance due are correct before sending them to third parties. And this is only suitable for businesses that do real-time recording.
I think the point missed by @Misu_Khan is from an accounting point of view an Invoice should only record that particular transaction detail for that specific transaction. A Statement from an accounting point of view records ALL outstanding invoices less any Payment made by that customer.
Combining the two distinctly separate document purposes into one combined document can only create issues within Manager and would not be recommended even if you can achieve this via scripts for the reasons already discussed.