How to get mongodump --quiet to still report errors?
As you already said 


  I've tried 

mongodump --quiet



As per mongodb documentation Here 


  Runs the mongodump in a quiet mode that attempts to limit the amount
  of output.


This option suppresses:

output from database commands
replication activity
connection accepted events
connection closed events


mongodump is an useful tool to backup Mongo database. Apart from taking a cold backup, it can also take hot backup. i.e You can connect to a running instance of MongoDB and take a backup even when users are using the database.
There are two way you can mongdodump


Backup by Shutting down Mongod Instance
Backup without Shutting down Mongod Instance


For further your ref from @RAMESH NATARAJAN Blog How to Backup and Restore MongoDB using mongodump and mongorestore