Years later and this is still a problem.
I have a Chromecast with Google TV (Ethernet/media hub with power pass through adapter), and I thought I'd found a solution which was to allow the settings app through NordVPN with split-tunnelling. Worked for a week and then stopped.
As soon as I connect to a server, I get the dreaded 'Ethernet not connected' and 'internet connection lost' messages.
Is there really no option other than to use WiFi? I just can't believe it at this point.
I forgot to mention I have heard of issues w/Android TV 12 so u may need to wait for an app update on Nord's behalf. sorry I missed that... I'm running 11 on my Shield so I'm guessing that's why I haven't had a problem yet. if I recall even IPvanish is working on a fix for the 12 OS as well... I'd say give it a lil time n u should be good. they're usually pretty quick to solve these things. if u get a chance email them the issue n if I have time I'll do it myself just bcuz... again g'luck n I hope it all works out asap.