Is there a way to add the template selection dropdown to the custom post type? Perhaps under Page Attributes?
So far it seems there is no native way to do it in WordPress, correct me if I am wrong. If you are okay with using plugins there is Custom Post Template that you could try.
Now in WP 4.7+ its easy ;)
According to https://make.wordpress.org/core/2016/11/03/post-type-templates-in-4-7/ it is enough to create a page template stating which Post Types to associate with in the template header comment:
/* Template Name: Full-width layout Template Post Type: post, page, product */
The dropdown to select the template will appear automatically in the 'Post Attributes' metabox.