That is very annoying. I remember having spent some time with this same issue in Google App Engine as well, which also runs Cloud SQL Proxy as a sidecar container.
Just FYI for people who don't know about it yet: with cloudsql-proxy v2 there's a new parameter called "--quitquitquit" that starts up an HTTP endpoint to be used for graceful shutdowns. Basically your main container makes a POST to this endpoint, and sidecar exits.
https://cloud.google.com/sql/docs/postgres/connect-kubernete...