Hiya Bridget (@Thingfully),
I've never used AirPlay, so I'm driving blind on this. But I would conclude the obvious - there is a conflict between AirPlay and and the cloudBit WiFi connection. My first attempt at a solution would be for separate WiFi networks. Separate connecting computers and routers as much as possible, and use different named networks.
You might be able to test this out by setting a phone to be a mobile hotspot for one of the connections The cloudBit would likely be the best choice here, as it will undoubtedly have lower bandwidth then the video channel being sent to the projector. If it hooks up this way, then that is indeed the culprit.
Perhaps littleBits' cloudBit team can look into any conflict with Apple's Airplay. Like you, this seems like it could be a very useful combination.