Issue Connecting to Cloud SQL Postgres using Private IP from GKE

I was able to connect to Cloud SQL Postgres by creating a VPC-native cluster as suggested by @patrick-w.

My VPC subnetwork creation was modified to include two secondary ranges:

gcloud compute networks subnets create stg-vpc-us-central1 \ --network stg-vpc \ --region us-central1 \ --range 10.10.0.0/16 \ --secondary-range stg-vpc-us-central1-pods=10.11.0.0/16,stg-vpc-us-central1-services=10.12.0.0/16

And my cluster creation command was modified to enable ip-alias, and added details of the secondary ranges to use.

gcloud -q container clusters create cluster-1 \ --zone us-central1-a \ --num-nodes 3 \ --enable-ip-alias \ --network stg-vpc \ --subnetwork stg-vpc-us-central1 \ --cluster-secondary-range-name stg-vpc-us-central1-pods \ --services-secondary-range-name stg-vpc-us-central1-services