I've still been chatting to Tutum support about getting this deployed in Docker (because I WON'T LET IT GO), and they suggested using the Tutum API, which was a real 'why didn't I think of this sooner' moment. This would get around the issue with volumes when we deploy this with Docker, and also would mean we don't need to mount the docker socket as a volume too.
This isn't necessary for the end of this sprint, but would be a nice improvement for the future, and would make things MUCH neater.