How many Apple TVs are there in the building? For debugging, run dns-sd -B _airplayin Terminal which will show the underlying Bonjour services. If they disappear and reappear often then there definitely is a problem.
John, Bonjour can be managed successfully, but there are things you can do to make it better. We have lots of Apple TVs, wired to a VLAN that is also bridged@AP to our Staff topology. I have IPv6 blocked (which halves the multicast traffic) and the Apple TVs have static IPs assigned through DHCP (which avoids problems with Bonjour Sleep Proxy), and are set to never sleep. I also drop port 5353 from Apple TV MAC addresses with a policy on our core S4 so devices in each building on see nearby Apple TVs.