How to grant permission to docker user write on /var/?
Have you tried via named volume (if using a docker version >= 1.9)?
$ sudo docker volume create --name confluenceVolume$ sudo docker run -v confluenceVolume:/var/atlassian/application-data/confluence --name="confluence" -d -p 8090:8090 -p 8091:8091 atlassian/confluence-server
Add ENTRYPOINT script and use it. Example - https://github.com/AfsmNGhr/dockemacs/blob/master/Dockerfile#L34. See also - http://www.johnzaccone.io/entrypoint-vs-cmd-back-to-basics/.