I ran a test on my system for you. I disconnected my Airport from the Internet yielding a wireless internal network only. With an existing ATV initially asleep, I was able to use the AirPlay function from an iPad to the ATV.
However when I pulled the power on that same ATV and rebooted it, the boot did not complete and it told me I had no network connectivity. It allowed access to settings and I had complete connection to the WiFi. That WiFi did not have connectivity to the network it really meant which was the Internet.
So unless you can boot the ATV with wifi and Internet then disconnect the Internet without rebooting the ATV, it will not work.