Can I fail a partially paid invoice?

Unfortunately, there is not a way to mark the invoice partially paid.  The best solution here would be to mark the current invoice paid and then create a new invoice ("Add Charge") to create a new invoice for customer reflecting the amount that is due.

If it was a manual invoice, you can record the partial payment and leave the remaining balance open and the invoice would remain open until its fully paid off. But for automatic invoice, we don't have this option as of yet.