OK, I've solved the issue for myself and am back to explain how to fix it in case anyone else comes across the same issue.

Firstly, you need to log into your original router (the one that the internet goes directly into) and change the DHCP settings so that your computer with the Plex server on it has a reserved IP address. The Powerline adapter gets the IP addresses assigned by your original router, so by forcing it to assign your computer the same IP address every time, the Powerline adapter stops changing the IP randomly and allows Plex to work. Remember to reboot your Powerline though.

Additionally, the issue was caused by trying to cast directly from the computer to the Chromecast. This is still in Beta and so isn't as stable as casting from an app. Trying to cast the same media from the app instead worked perfectly, whereas it still has trouble from the PC.

Hope this helps someone!