I finally managed to get Transmission-daemon running inside the Docker container through an OpenVPN connection to Privateinternetaccess.com.
Here is the Dockerfile and the various configuration files and scripts needed to make this work. I'm open to any simplifications and/or alternate ways of achieving this! Some notes:
OpenVPN requires running the container with the --privileged flag in order to create the tun0 connection.
I could only make it work right by using Pipework along with a bridge on the host to give the container it's own IP address on the network.