install wp-cli and then use the wp-cli commands provided by the plugin to clear the cache via the command line, and then you can set the command as a cron job
the command may look like this: wp --path=<path to wp doc root> w3-total-cache flush
you need to specify the path so that it knows which path to operate on as wp expects by default to work on the install found directly in the path you are in, which may not be the same as the home directory for the user that runs the cron task
also try making sure you have enabled category page purging as well in the page cache config