Automated docker builds with private bitbucket server
I would use a Packer template, specifically with a Docker builder to do this. Triggering this build can be done with hooks (either BitBucket webhooks, or git server post-receive hooks.

Building a Docker image from a Packer template has some benefits, but requires Docker Engine installed - so, doing the build with a Dockerfile would also be an option.

The tricky part would be the management of credentials. Building the image does not require credentials, nor does tagging -- however you need to be authenticated to push the image to the registry. If the build happens on the same server that is hosting BitBucket, you could keep them in a protected file and execute the build via the post-receive hook in a shell script. This would be a lightweight option, especially if the server itself is locked down.

So, in summary : 


Install Docker Engine on the BitBucket Server
Write a script which can consume registry username and password, to build the image, either from a Dockerfile or from a Packer template.
Add a server-side git hook to execute this script to build, tag and push the image.