Thanks Paul!

I think the minimum capacity for Aurora serverless still exists. It says the minimum is 0.5 and the rate is 0.06/hour for half a unit. That's about $40 just for the server time, which seems overkill for a low traffic app.

If you're okay with slower performance and a complicated deployment, you could host your own SQL server on a small EC2 instance. If you're doing this, you might aswell get lower latency by hosting the backend server and SQL server on the same EC2 machine.

If you don't have to stay within AWS, there seems to be non-AWS cloud providers that seem to offer low throughput of SQL for free: https://developers.cloudflare.com/d1/platform/pricing/.