increasing disk space for mysql data volume
You say you have bin logging enabled.  This leads me to believe you have replication running.

To minimize down time you could promote your slave to master.  The finer details on how to do this really depends on your setup.  Maybe it will involve updating application configurations, maybe it will involve changing the DNS to your "master server".

Prior to the promotion ensure you slave is configured they way you would like, e.g. not writing binlogs to the data directory.  Make sure all the grants present on the master are on the slave.  Make sure the slave is not running in read only mode when you are ready to bounce.

Prior to promotion you'll want to ensure both the master and slave are in read only mode.   Run show master status; on the slave to see what log file and position you'll use for change master to after you reconfigure and restart the old master to run as a slave.

Once the old slave is promoted to the new master you can take your time to move the old binlogs to a new file system and reconfigure to keep new logs out of the data dir.