syntax error, unexpected '[' in brightcove.install

Assuming that by "Php 7.6.30" you're referring to php 5.6.30 and by "correct ownership" you mean enabling all permissions on the API - I wasn't able to reproduce your issue within this docker container: https://github.com/janmashat/docker-drupal7-brightcove6

The docker container defaults to the stable version of the brightcove module, but you had indicated 7.x-6.x-dev in the issue metadata - so I made this modification (before running docker build) to grab the latest dev release:

docker-drupal7-brightcove6 (master)$ git diff
diff --git a/start.sh b/start.sh
index 0210d36..b22e1dc 100644
--- a/start.sh
+++ b/start.sh
@@ -46,6 +46,7 @@ if [ ! -f /var/www/html/sites/default/settings.php ]; then

        # Download Date (required: date_popup, date_api)
        drush dl date
+       drush dl brightcove-7.x-6.x-dev

        # Install PHP-API-Wrapper
        cd /var/www/html/sites/all/libraries

I ran the first cache-clear before adding an API client, and the 2nd one after adding it:

root@c20377f773aa:/var/www/html# drush cc all
'all' cache was cleared.                                                      [success]

root@c20377f773aa:/var/www/html# drush pm-list |grep brightcove
 Brightcove        Brightcove media integration (brightcove_media)              Module  Not installed  7.x-6.4+4-dev
 Brightcove        Brightcove Video Connect (brightcove)                        Module  Enabled        7.x-6.4+4-dev

root@c20377f773aa:/var/www/html# drush cc all
'all' cache was cleared.                                                      [success]

root@c20377f773aa:/var/www/html# php -v
PHP 5.6.30-0+deb8u1 (cli) (built: Feb  8 2017 08:50:21)
Copyright (c) 1997-2016 The PHP Group
Zend Engine v2.6.0, Copyright (c) 1998-2016 Zend Technologies
    with Zend OPcache v7.0.6-dev, Copyright (c) 1999-2016, by Zend Technologies

root@c20377f773aa:/var/www/html# drush status
 Drupal version                  :  7.54
 Site URI                        :  http://default
 Database driver                 :  mysql
 Database hostname               :  localhost
 Database port                   :  3306
 Database username               :  drupal
 Database name                   :  drupal
 Database                        :  Connected
 Drupal bootstrap                :  Successful
 Drupal user                     :
 Default theme                   :  bartik
 Administration theme            :  seven
 PHP executable                  :  /usr/bin/php
 PHP configuration               :  /etc/php5/cli/php.ini
 PHP OS                          :  Linux
 Drush script                    :  /root/.composer/vendor/drush/drush/drush.php
 Drush version                   :  8.1.11
 Drush temp directory            :  /tmp
 Drush configuration             :
 Drush alias files               :
 Install profile                 :  standard
 Drupal root                     :  /var/www/html
 Drupal Settings File            :  sites/default/settings.php
 Site path                       :  sites/default
 File directory path             :  sites/default/files
 Temporary file directory path   :  /tmp

Does that mean my assumptions above were correct?
Maybe you can try with drush 8.1.11 to see if that's the culprit.

Drush 8 should be compatible with Drupal 6 according to http://docs.drush.org/en/master/install/