Connect a Google CloudSQL Postgres database to Data Studio

Eventually it turns out that just changing the format of the client key works. The reason the above didn't work was that the command to change the format was wrong.

It should be:

openssl pkcs8 -topk8 -inform PEM -outform PEM -in client-key.pem -out client-key.key -nocrypt

(notice the PEM instead of DER parameter that is posted in the question).

So Data Studio connection works! Happy days!