Docker Hub automated builds and Travis CI
If you're married to Docker Hub, you could use a webhook to notify Travis of the need to start it's job.

However, I solve this problem using Quay.io - it has the robot account (you refer to "service account") you need. It also has a richer set of notifications.

You could maintain your current workflow using webhooks to 


create the repository on Quay or Docker Hub, have the image built on git events.
Create a webhook in the repository to send data to your Travis job when builds succeed


Or, you could build your images in the Travis job.


Push the initial image to a repository on Quay
Add a robot account to your repository in Quay
Add the necessary secrets to your Travis build (token and robot user name)
Configure Travis to build the image, run whatever tests you need and then push on green.
Add an after_success task to redeploy your app in k8s after the push.