Simplest filesystem backup of Postgres database
Yes, base/ will likely be changed while the backup is occurring.  And yes, you need all of the WAL segments for the time between pg_start_backup() and pg_stop_backup().

Your existing filesystem backup tools do not provide you with the tools you need to back up a live database.  This is not unusual.  Rather than trying to fit a square peg into a round hole, why not just tell your filesystem backup tool to ignore the database, and then use a dedicated database backup tool to backup the database?  For example, use pg_basebackup in conjunction with WAL archiving, or pg_basebackup -X s as a standalone backup.  Of course, the files created as the result of this backup need to themselves backed up, but at that point they are "cold" files and your regular backup system will work.