MQTT client and HTTP module don't work together?

So you're saying that if you just run require("MQTT").create on its own, without the AP, it works?

Can you try doing something like:

apServer.close()
setTimeout(function() {
  var client = require("MQTT").create(server, options /*optional*/);
  client.connect();
}, 2000);

And see if that works ok?

The calls to the ESP8266 will be asynchronous, so I guess it's possible that the driver isn't correctly delaying the creation of the new MQTT connection, and so is trying to re-use the apServer socket that hasn't yet closed?