Architecture: net.connect() and connection callbacks ...

The blocking connect works in this case because the library is sending a Serial request and can then happily block waiting for a serial response

No, it doesn't block at all. Your implementation just needs to do:

bool isBusy = false;

function esp8266_connected() { isBusy = false; }
function esp8266_sent() { isBusy = false; }
function esp8266_received() { isBusy = false; }

function connect() {
  esp8266_connect(...);
 isBusy = true;
}

function send() {
  if (isBusy) return 0;
  isBusy = true;
  esp8266_send(...);
  return bytes_sent;
}

function recv() {
  if (isBusy) return 0;
  isBusy = true;
  esp8266_recv(...);
  return bytes_received;
}