I have had this issue before, but it was a little different. I would playback a recording and then after a while of playing it would cut off and give me the disconnect message. I'm not sure what I did really to fix it, as I followed your instructions to set mine up. Do you have the backend/switch set to manual from your router instead of DHCP? Not sure that would be causing an issue but just a thought.
The other thing I did was buy one of these for my backend: https://www.amazon.com/TRENDnet-Ethernet-Chromebook-Specific-TU3-ETG/dp/B00FFJ0RKE/ref=sr_1_2?ie=UTF8&qid=1490277519&sr=8-2&keywords=trendnet+usb+ethernet
I read into a lot about how the ethernet connection on all the Pis is throttled because it shares a bus, and that ethernet adapter is full USB 3.0 and supports gigabit. People ran tests and it had much better connection speeds through the adapter than the built-in ethernet.