Docker: LetsEncrypt for development of "Https everywhere"
I suggest you forget about Letsencrypt. The value proposition of that service is really focused on "getting that green lock in the browser", which you explicitly say you don't require.
Also, Letsencrypt requires access to your server to verify that the ACME challenge file is there, which means YES, you need every such server to have a publicly reachable domain. So you need to own the domain and have DNS pointing to your specific server, which sounds undesirable in a testing environment.
So in summary I think you're trying to use the wrong tool for your needs. Try using regular self-signed certificates as described in this question. For that to work, the connecting clients must be set to not verify the certificates.
Or you can take it to the next level and create your own CA. For that to work, you need to make all your containers import that root cert so that they will trust it.
Of course, once you ship the containers/images into production, don't forget to undo these things and get real valid certificates. That's when Letsencrypt will be useful.