So I might be putting words in your mouth, so please correct me if this is wrong. It seems like you don’t actually control the SIGTERM handler code. Otherwise you could just write something like:

  sigterm_handler() {
    make_healthcheck_fail();
    sleep(20);
    stop_web_server();
    exit(0);
  }

Technically the server shutdown at the end doesn’t even need to be graceful in this case.