Why not cancel the new invoice and pay the old one as they said?
Why not? You do have an idea about them: TeamViewer asked you to pay them.
> I suspect they want me to do this because closing internally with the company will mean admitting to the agency that they sold a bad debt.
Maybe. Maybe that's just how it's easier for them.
This screams of dodginess and shoddy unfriendly customer service.