ApiServer.scala

class ApiServer extends HttpApp {

  protected val design: Design = bind[Design] // ApiServerの外部で定義されているDesginを参照したい
    .bind[AtomicReference[ActorSystem]]
    .toInstance(systemReference)
    .bind[ActorSystem]
    .toInstanceProvider[AtomicReference[ActorSystem]](_.get)
  
  // snip
  
  override protected def routes: Route = bind[Routes].route
  
}

Airframe doesn't support binding inside classes, so you need to use constructor injection:

class ApiServer(design:Design)

or trait injection:

trait ApiServer {
   protected val design: Design = bind[Design]
      .bind[X].toXXX  
}

結局、HttpAppは抽象クラスで使いにくかったので利用しないで、traitで書きました。コメントありがとうございました。

trait ApiServer {

  implicit val system           = bind[ActorSystem]
  implicit val materializer     = ActorMaterializer()
  implicit val executionContext = system.dispatcher

  private val routes = bind[Routes].routes

  def start(host: String, port: Int): Future[ServerBinding] = {
    val bindingFuture = Http().bindAndHandle(handler = routes, interface = host, port = port)
    sys.addShutdownHook {
      bindingFuture
        .flatMap(_.unbind())
        .onComplete { _ =>
          materializer.shutdown()
          system.terminate()
        }
    }
    bindingFuture
  }

}