Connecting with TLS/SSL certificates without the Cloud SQL proxy is definitely supported.
I can't say for sure without more details, but it sounds like your driver is trying to fully-verify the certificate by making sure that the CN resolves to the same IP you are trying to correct with. I don't know which database engine you are using, but
here is the postgres docs that explains this behavior:
> Once a chain of trust has been established, there are two ways for the client to validate the leaf certificate sent by the server. If the parameter sslmode is set to verify-ca, libpq will verify that the server is trustworthy by checking the certificate chain up to the root certificate stored on the client. If sslmode is set to verify-full, libpq will
also verify that the server host name matches the name stored in the server certificate. The SSL connection will fail if the server certificate cannot be verified. verify-full is recommended in most security-sensitive environments.
If you make sure your driver is using `sslmode=required` or `sslmode=verify-ca`, you should be able to connect without any issues. Here's an example configuring a
TCP connection using TLS.Hope that helps.