Sounds like just setting show_ui to false will do what you’re after – it will hide the taxonomy in the admin menu, and it won’t create a metabox on the post edit page.
After this:
'hierarchical' => true,
Add this:
'show_ui' => false,
You’ll find a full reference of all the available arguments over at the WordPress codex: https://codex.wordpress.org/Function_Reference/register_taxonomy
EDIT: (thanks to Howdy_McGee in the comments)
This will hide from all backend users including admins. If you want the taxonomy to be visible just for admins but not lower level users, instead use:
'show_ui' => current_user_can( 'administrator' ),
Instead of ‘administrator’ here, you can also use any role at https://codex.wordpress.org/Roles_and_Capabilities if you want more fine-grained control.