Essentially it comes down to what a few people have said.
Secrets should be managed outside of a docker container and passed in via Volume Mounts or via Environmental Variables.
My suggestion would be to have the docker host systems manage let's encrypt and mount the let's encrypt as a volume inside your container. That way you could have multiple containers using one current SSL certificate instead of trying to maintain multiple instance of let's encrypt and certificates.
This should help with the too main requests for a given domain issue.
have a look at this one SSL for Docker Containers Showing error