I can AirPlay a still screen from my MacBook Air to my Apple TV but I can't AirPlay video. The audio comes over but not the video.

Not all content will work with airplay, it is up to the developer to allow it (and some make the choice to block it). If it is stated to work then it is a glitch that they need to address. If you want to be sure you can contact them directly.

There may be a FAQ section that denotes whether it supports airplay, otherwise there should be a contact method for which you can use to ask them directly.