The ability to override the background currently exists in only the “Post” and “Page” post types (in the form of an additional “Meta” box called “Background” visible on each Post or Page edit screen).
This can be changed manually, by editing the file (...)/wp-content/plugins/background-manager/app/Myatu/WordPress/BackgroundManager/Meta/Single.php
on line 28:
protected $pages = array('post', 'page');
Simply use the registered name of the Custom Post Type.
If you wish to have the ability to do this from within your own plugin, or via the Background Manager settings screen, let me know. It would be possible to add a filter for this purpose.
Edit: The same goes for activation/deactivation on specific post types, however, this would require slightly more manual editing at the moment. Namely (...)/wp-content/plugins/background-manager/resources/views/settings.html.twig
and various sections of code within (...)/wp-content/plugins/background-manager/app/Myatu/WordPress/BackgroundManager/Main.php