Hello Joe,
Following this tutorial, the container containing your application, and the Cloud SQL proxy container (sidecar) should be tightly coupled in a single manageable entity (Pod) per your deployment (wordpress example), and the same applies for each replica pod. This is the recommended method to use with Kubernetes. But, I am not sure if it would be ideal for your use case. That said, and as you mentioned you may test deploying Cloud SQL proxy as a cluster service using Helm per this Github example
To answer your last question about the username, the tutorial for clarity reasons, and for easier cleanup ( when you finish the example) suggests to create a new username. It is not necessary if you already have a username/password you plan to use. I hope that helps.