@faraday It turns out not to be too terribly difficult, at least at the level I'm working at right now. There's certainly more advanced Docker stuff that will probably result in images that are more secure and which do things 'properly' but for now what I'm doing seems to work.
What I did was I created a Docker instance for Ubuntu (I actually tried some more minimal setups at first but went back to Ubuntu because of some of the tools I wanted). Once you spin that up you can log into it using a command similar to docker exec -i -t mocker_game_1 /bin/bash (mocker_game_1 is the name of the Docker instance I'm running and /bin/bash gives me a shell inside of it).
From there I basically figured out all the steps that I would have to run to setup a new tinyMux server (downloading build tools, configuring, compiling, etc.). You then create a Dockerfile (not to be confused with docker-compose.yml) that repeats those instructions. You run a new command along the lines of docker build -ttinymux:2.10-U ~/Docker/mocker/release/ which tells Docker to build a new image called tinymux:2.10-U using the Dockerfile found in ~/Docker/mocker/release/.
That's sort of the 'elevator pitch' version of it, anyway. It does get a little tricky since the commands in the file can't pause for input and you need to work in stuff so that when you run things for the very first time data gets moved around so it is accessible from outside docker (for persistence and so you can back it up) but if you're already capable of compiling a game on a new machine you can probably handle those extra issues.
Like I said, I will provide more complete details about everything I was doing so that you can build your own Arcker (or whatever you'd like to call it) image. I just want to wait a day or two in case someone says 'oh, Holy Hell, the way you've done this will lead to the eventual extinction of life on Earth. Don't do it that way' or something similar before I write out the more detailed account. This is just to show you that it is probably not hard at all (having never spun up a copy of Ares I don't want to absolutely promise it will be easy since there might be some gotcha I don't know about) for you to build your own image.