Solution:
The setting for admin bar option is saved within the wp_options table with the key "toolset_options" where we save the settings as serialized array.
You just need to get the value of key "toolset_options" from wp_options table and then unserialize it and look for the array key "show_admin_bar_shortcut" which holds the value either "on" or "off".
If you want to display the "Design with Toolset" menu, you should update the "show_admin_bar_shortcut" to "on" and if you want to disable it then set it value to "off" and accordingly save the "toolset_options" key value.
