I eventually came up with a solution that works for me and greatly improved my Docker experience.
Long story short: I used a combination of Fabric and a container running sshd.
Details:
The idea is to spawn container(s) with sshd running using Fabric's local, and run commands on the containers using Fabric's run.
To give a (Python) example, you might have a Container class with:
1) a method to locally spawn a new container with sshd up and running, e.g.
local('docker run -d -p 22 your/image /usr/sbin/sshd -D')
2) set the env parameters needed by Fabric to connect to the running container - check Fabric's tutorial for more on this
3) write your methods to run everything you want in the container exploiting Fabric's run, e.g.
run('uname -on')
Oh, and if you like Ruby better you can achieve the same using Capistrano.
Thanks to @qkrijger (+1'd) for putting me on the right track :)