Docker consumer on AWS while using RabbitMQ

I have most of my app designed and deployed. We have gone for RabbitMQ over SQS for several reasons.

I currently have my consumer running in a docker container and would like to deploy this on AWS. I am fairly familiar with Elastic Beanstalk since our web-tier is running there, but it seems all workers deployed this way have to use SQS?

The other option I am aware of is to use ECS for the docker component, but I do not want to make the image publicly available and don't have access to a private repository.

Is there some basic functionality I am missing, a document describing how to deploy to ECS using a Dockerfile and source code locally, or a way of deploying to EB using a Dockerfile without being locked into SQS?