Big ups for calling out how bad an idea connecting to the database is from a serverless context. I do a lot with serverless on AWS Lambda. Since part of that involves an ETL to write to a database, I also run a server that exposes a service for writing data via RPC. The server provides a connection pool to the database and appropriately encapsulates all the complex functionality we have for incoming data behind a well-defined interface.

If you really insist on calling your entire stack serverless, do what we do and run these servers as ECS tasks via Fargate. That is plausibly serverless, albeit long-running. You get all the perks of a serverless environment with all the perks of something like beanstalk (without having to patch a server!). The drawback in this scenario is that running ECS tasks via Fargate don't provide as much flexibility in cost.