using wp enqueue style to create a CSS file specifically for a page template

If “/styledtwentyfifteen” is your child directory then that’s included by get_stylesheet_directory_uri() (of which bloginfo('stylesheet_directory') is an alias) so you should leave it out on registering:

function register_more_stylesheets() {
    wp_register_style( 'stylesheet_name', get_stylesheet_directory_uri() . '/background-slider-template.css' );
}

Similarly with is_page_template() you only need the file name (relative to the template directory), and if you register a style you should just use its handle when enqueuing:

function add_my_stylesheet() {
    if ( is_page_template( 'background-slider-template.php' ) ) { // using page slug
        wp_enqueue_style( 'stylesheet_name' );
    }
}