AppleTV randomly displaying Wifi Not Connected when using Ethernet

I don’t think that message is from Apple TV. Instead, it’s probably from the TV itself.

Apple TV notifications are less transparent, floating more to the right, have much rounder corners, a bit smaller type, different grammar (would omit “is”).

You could connect the TV to ethernet too, if it supports that.

Else contact Sony for advise.

Also see Network Diagnosis from Sony, depending own your TV’s OS.