Does anyone know of some SQL clients that can connect to Aurora Serverless? I'd like to visualize my whole database and the query editor in the RDS console is pretty shit.

If you setup an ssh tunnel many clients can talk to the aurora serverless instance, e.g. DBeaver. But directly supported, I'm not sure.

Yeah, I’m not aware of clients that can connect through the Data API. But you can still connect to it normally if the SQL client access the RDS server (ie. VPN or publicly exposing the instance)