Insert Widget option into mark-up with register_sidebar

First you will replace this line <div id="%1$s" class="widget %2$s" data-custom-title=""> with <div id="%1$s" class="widget %2$s" data-custom-title="DCT">

Now add the filter in your functions.php file.

add_filter( 'widget_display_callback', 'add_custom_data_to_before_widget', 10, 3 );
function add_custom_data_to_before_widget( $instance, $widget_class, $args ) {
    if ( ! empty( $instance['title'] ) && ! empty( $instance['custom-title'] )) {
        $args['before_widget'] = str_replace('DCT', sanitize_text_field ( $instance['custom-title'] ), $args['before_widget']);
        $widget_class->widget( $args, $instance );
        return false;
    }
}

Hope that this will help you.