You should persist certs somewhere. Otherwise your availability is heavily tied to LE’s uptime.

Technically, because Let's Encrypt always publishes all requested certificates to the logs (this isn't mandatory, it's just easier for most people so Let's Encrypt always does this) your tool can go look in the logs to get the certificate. You do need to know your private key, nobody else ever knew that so if you don't have that then you're done.