I need to completely ‘wp_dequeue_script’; what’s the best way?

Your logic is basically correct. You’re just missing a ' in your wp_dequeue_style() function.

Also, wp_dequeue_style() only accepts one argument: the handle of the script/style you want to dequeue. You don’t need to pass the url. So your code would need to look like:

if ( is_page_template( 'specific-template.php' ) ) {
    //  Remove Site-wide CSS
    wp_dequeue_style( 'generic styles' );
    // And this would enqueue the unique css
    wp_enqueue_style( 'unque to this template css', get_template_directory_uri() . '/css/unique-to-this-template.css', array(), '1.0');
}

You’ll also need to make sure that this code is hooked after the original style is enqueued.

If this is all your CSS from the same theme though (and not a child theme), then I’d suggest a different approach: Rather than dequeueing a specific style, just selectively enqueue it:

if ( is_page_template( 'specific-template.php' ) ) {
    wp_enqueue_style( 'unque to this template css', get_template_directory_uri() . '/css/unique-to-this-template.css', array(), '1.0' );
} else {
    wp_enqueue_style( 'generic styles', get_stylesheet_uri(), array(), '1.2' );
}