How do I add Aurora serverless cluster endpoints to Service Connect using CDK?

I am planning to use AWS Service Connect for backend communications between my micro-services to avoid the costs of an internal load balancer.

I figured out how to configure my Fargate services for Service Connect using service.enableServiceConnect(...).

However I can't figure out how to publish the endpoints of my Aurora serverless cluster to Service Connect? What I want is tasks in the cluster being able to connect to Aurora using a well-known internal DNS name (rather than passing . The cluster is created like this:

  this.dbCluster = new rds.DatabaseCluster(this, 'DbCluster', {
            engine: rds.DatabaseClusterEngine.auroraPostgres({
                version: rds.AuroraPostgresEngineVersion.VER_15_2,
            }),
            instances: 1,
...

Thanks!