Allow shortcode for custom widget

You can try the following:

/**
 * Only allow shortcodes for a widget with a given title.
 *
 * @see http://wordpress.stackexchange.com/a/160246/26350
 */

function wpse_widget_shortcode( $text, $instance )
{
        // Only allow shortcodes for the widget with the following title:
        $title="Some Title"; 

        remove_filter( current_filter(), __FUNCTION__ );

        if( isset( $instance['title'] ) 
            && mb_strtolower( $title ) === $instance['title'] 
        )
            $text = do_shortcode( $text );

        return $text;
}

add_filter( 'widget_text', 'wpse_widget_shortcode', 99, 2 );

where we target the given widget by it’s title.