iTunes, rented movies, and AirPlay to appletv

Hi,
I had an issue with iTunes today that sounds strange. So I normally rent iTunes movies either from my iPad or from my living room appletv.

However today we wanted to start watching a movie in the kitchen appletv and then continue in the living room. Therefore I thought the best way would be to rent it from my mac from iTunes, and AirPlay mirror my mac to my kitchen appletv and then moved into the living room appletv.

I found however that once I started to AirPlay mirror my mac, iTunes would only show black and grey squares instead of the video.

I checked but I see no obvious way to stream from iTunes to the appletv directly without using Mac OS X AirPlay mirroring.

Am I missing something ???
What would you have done differently??

Thanks!!