sounds like a good idea. I'll try it.
Ya but I'm thinking you can generate a new account using the subdomain (mail.yourdomain.com) as the account domain on your mail server. It will need an A record pointed correctly to it and Lets Encrypt should issue a new SSL for it when you click create (just ensure AutoSSL is ticked under additional options). Lets Encrypt certs can't do what you want at the moment but it might work for a subdomain.
I haven't tested but am assuming this might work for you.