Pay for Secret Manager. Those hardware security modules and compliance attestations to implement it are not free.
Or come up with a different solution. Different secret management software, less sophisticated store secrets in plain text files, different application deployment model. Which may not be compatible with exclusively using the managed Cloud Run product.
unclear to me how much of a security risk that actually poses given
that the project is a student website that doesn't contain any
sensitive personal information in the database
Possibly a small risk to your organization in the big picture, but being careless with secrets is a bad habit.
Every application, even small ones in a simple compliance environment, deserves some level of protection of secrets. At least restricted access configuration files. Definitely not embedded into application code.
a user with administrative access to the Google Cloud console could
access the secret values from secret manager anyway.
Hiding data from authorized administrators is not the point of secret systems. Any access or deletion of secrets is logged. Which in theory enables catching disclosing these to an attacker, or use of old versions beyond the time when they were supposed to be rotated.