I have contacted the AWS Support folks regarding the same problem. Their response is as follows:

There is a much simpler solution by running user-data. As far as the code running inside a container is concerned, there isn't a difference between containers or VM's. The code thinks its running on a regular OS. So instead of creating the docker container with the Dockerfile, you can use a user-data script to do the same things on an instance. For example, with ADD in a Dockerfile, it takes files and writes them to the container. We can pull this file from, say S3, and copy it wherever on the instance is needs to go. It will go in the same place that it lives in the container. The RUN directives in a docker file will map 1-to-1 with a user-data script, since these are simply commands. For the CMD directive, we can simply run that process via user-data. Docker volumes are irrelevant, since we have access to the full storage of the instance, so you can ignore the creation of volumes and simply write wherever any files need to go. In summary, your user-data script will replace the Dockerfile for bootstrapping your instance and running your application. Instead of Dockerfile syntax, you will use Bash syntax. Look below for an example script that mimics your Dockerfile.