Question: What solution can we use to send a PDF invoice by mail to each subscription and on each renewal?

I'd suggest the use of s2Member's event-driven API Notifications for this. You can connect an email address or URL (Webhook) to every payment that s2Member processes and/or receives notice about; i.e. for every initial payment, and for every recurring payment too. This would allow you to generate a PDF invoice using a 3rd-party invoicing service, or through a custom script that you develop on your own.

s2Member itself does not support the automatic generation of PDF files.

Please see: s2Member → API / Notifications → Payment Notification
See also: Building an API Notification Handler (Webhook)