Why do you need Docker to monitor it? I would think it will only recognize a problem if the process dies. But you need to monitor your db and application for other types of failures anyhow, right? I cannot remember the last time I had a problem with the DB process dying.
That's default behavior. Docker tracks the main process. If it dies, and your database is not the main process, then the entire container will be restarted (depends on configuration). Means your database will be forcefully killed. At some point you might run into data corruption incident.
If you make database your main process, then docker will be blind for your application death.
That's why I mentioned that if you want to put multiple applications inside a single container, you have to put it under supervisord or alternative. So neither would be the main process.