How do I play an episode from my iphone on Apple TV while Apple TV has it partially downloaded?

If it's loaded on the iPhone you simply bring up control centre to enable AirPlay and then start playing on the phone, which will then stream to the ATV

Just to clarify, when you AirPlay video on a server from iPhone to ATV, it does not actually transmit the video over the WiFi. Only the stream identifier is sent to ATV, which connects to the server and streams it directly. You can check this by AirPlaying a video from Safari on the phone. You will not see the downloading wheel on the phone because ATV is downloading directly. If you enable AirPlay mirroring then, of course, you will be sending the data direct from phone to ATV, but you won't have full-screen video. Also, if you play a video from the iOS Videos app, it will be streamed direct from the phone, but you cannot do that with copy-protected video. To conclude, you cannot do what you wish because AirPlay doesn't work that way.

This business or how AirPlay works with a server is not widely documented. The clearest description is

http://www.flatpanelshd.com/focus.php?subaction=showfull&id=1347605891

which describes all the AirPlay modes.