Serverless Containers is first of all a tradeoff between running costs and maintenance costs. Besides the technical limitations, the financial aspect should be considered as well. A fully managed / Serverless Compute transforms the varied future costs of unknown maintenance, to a known fixed cost of zero maintenance paid for for now. Money invested into a system’s resilience. The price difference between running a Servereless Container and directly on EC2 is what I call a premium.
If your application requires persistent local storage, one that does not get deleted when a container is stopped, you can not go Serverless/Fargate. That is not even a tradeoff, that is a limit. You'd need to consider the alternatives of a Local EBS, a Remote EFS or a Remote S3.
You should wonder if a Container Orchestrator, a managed or unmanaged one, is of actual use. You should also wonder if and when should Containers not be used at all.
A database failure or a bottleneck is harder to resolve (failover, tuning or scaling) than a stateful/stateless application. A database is also a more “sensitive” component in the system. For both these reasons, the answer for should you use a container is it depends.