As it turns out, the problem is that Dovecot—which handles IMAP duties on the server—doesn’t notice when the certificate has been updated on disk; it will cheerfully keep using an in-memory cached copy of whatever certificate was present when the service started until time immemorial. //

I created a new root cron job to restart Dovecot once every Sunday at midnight:

# m h dom mon dow   command
0 0 *   *   Sun   /etc/init.d/dovecot restart

Since Certbot renews any certificate with 30 days or less until expiration, and the Sunday restart will pick up new certificates within 7 days of their deployment, we should be fine with this simple brute-force approach