Hi,
I have the same problem than you with the latest CC3000 version wich include all TCP/IP tool.
I have exactly the same logic result than we can see in this thread.
Except than with the evaluation TiWi-SL kit, the IRQ go to low state again. So, it's ready to send a response. This event occur approximatively 580ms after the first message sent to the module.
To conclude, the problem come from module hardware or firmware and not from your host firmware/software.
I hope that it can help you and i am searching some response.
Best regards.