Incorrect client response when database on Postgres instance doesn't exist

[w3b6x9]

Bug report

• [x] I confirm this is a bug with Supabase, not with my own application.
• [x] I confirm I have searched the Docs, GitHub Discussions, and Discord.

Describe the bug

Supavisor will continuously attempt to connect to a database even when that database doesn't exist. However, the client doesn't get the proper error feedback that database doesn't exist. See screen recording:

https://github.com/supabase/supavisor/assets/5532241/b6a15876-06ee-40b6-9e1b-e071025f68cd

To Reproduce

Steps to reproduce the behavior, please provide code snippets or a repository:

1. psql into a database that doesn't exist.

Expected behavior

Some sort of client error that clearly spells out there was an issue connecting to a database that doesn't exist.

Supavisor needs to stop doing additional work (following is repeated over and over):

16:22:27.870 [error] Error auth response ["SFATAL", "VFATAL", "C3D000", "Mdatabase \"bad\" does not exist", "Fpostinit.c", "L885", "RInitPostgres"] file=lib/supavisor/db_handler.ex line=227 pid=<0.987.0> project=dev_tenant user=postgres mode=transaction
16:22:27.870 [error] Connection closed when state was authentication file=lib/supavisor/db_handler.ex line=348 pid=<0.984.0> project=dev_tenant user=postgres mode=transaction
16:22:27.870 [error] Connection closed when state was authentication file=lib/supavisor/db_handler.ex line=348 pid=<0.987.0> project=dev_tenant user=postgres mode=transaction