If I understand your question correctly, you need option to select page template in a custom post type. In order to do that you have to take a different approach.
In your page-fullwidth.php add template name using the following code:
/*
Template Name: Full-width layout
Template Post Type: l
*/
get_header();
In theory, the default template should fall back to single-l.php but if not please do let me know.
Also to clear out the misunderstanding on page_attributes argument, it only provides option for selecting menu order and selecting a parent if hierarchical is set to true.