I don't use AirPlay so this is a shot in the dark, but from what I am reading it sounds like your iPhone can either connect to a device via WiFi directly or stream to it over the WiFi network. How is your Raspberry PI attached to the network, hardwire ethernet or WiFi? Can you direct connect the PI to the network via ethernet and just AirPlay to the WiFi? I wonder it there is just too much WiFi traffic and if pulling the PI over to a hardwired connection would help?
https://en.wikipedia.org/wiki/AirPlay