How to create additional rendering for custom post types?

Question

You need to use add_rewrite_tag and add_rewrite_rule functions to first register your custom URL handlers.
Something like

add_rewrite_tag( '%custom_render%', 'true' );
add_rewrite_rule(
        '([^/]+)/custom_render/?$',
        'index.php?name=$matches[1]&custom_render=true',
            'top'
    );

And then you can us template_inclide hook to check if the query param custom_render is set, then return the name of the new template.

Note: You need to flush your rewrite rules after making changes.