Today I decided to try something else with the Apple TV. I set up another Apple ID with iTunes store account. I then family shared my normal Apple ID to that.
I logged in as the new Apple ID and played episodes via the family sharing.
First episode fine.
Next episode - wants my iTunes password before playing...
And before someone asks, I also tried fixed IP config IPv4 only already.
Next step: physically swapping for another Apple TV
Update: didn't help - Apple TV using my account asked for iTunes password on playing one of the episodes. Arrrrg!