Without knowing the full details of your requirement, it seems an if/then statement could be integrated into the Invoice Reminder email that dispatches to the client (if configured in Automation Settings).
If Iām not mistaken, the payment reminder emails include support for product ID merge fields. This could be a simplistic solution using already in-place configuration/automation settings. You would just if/then and run a php script that sends the mail and possibly inserts merge fields from the existing template.