You could try leaving WiFi turned on, and then connecting via Ethernet, but I can't recall right now whether doing so automatically switches WiFi on. I'm hoping that if WiFi is left on, even if not connected to a network, that may be enough for the BBC iPlayer Android app to be happy.
If not, you could use the iPlayer add-on in XBMC, as I'm fairly sure it doesn't care how you're connected. You could also send a message to the BBC asking them to alter this behaviour in their app, but people have complained about it for a long time without much hope of a change.