I don’t feel like I’m certain in my answer, because I don’t understand your set up (and I’m not expert in this). BUT that said, let me explain further.
My environment has a robust local mail system. My environment has countless small VMs that are all pre-configured to have local mail forward appropriately.
When I set up Discourse, I installed (ran discourse-setup) on web-forum-1.$colo_name.$dotcom and configured it to use mailhost.internal.$dotcom for SMTP host and 25 for SMTP port, with no username or password. But it would have worked to use web-forum-1.$colo_name.$dotcom, too, because telnet localhost 25 works on that box, and Postfix there is configured to relay through mailhost.internal.$dotcom. That is a CNAME for one of the Exchange (eww, ick, says my inner Unix guru) servers, which does spam checks and routing to internal or external addresses.
(Separately I have the network configured to map the external hostname of discussion.$dotcom to the internal server web-forum-1.$colo_name.$dotcom because I want the service name to be independent of the backend.)
So going back to your question, since it seems you have a working Exim set up on your linux box, the Docker container should probably be configured to talk to that local Exim instance. The hostname will be the name of the “linux box running Exim” and the port 25, or 587, or one of the other two. And no authentication values.