I wish they would just start making (cheap ) 22/24” 4K TVs for second rooms. Would solve a lot of problems.
I’m semi confident the baluns will work as the kitchen tv isn’t involved in the audio decoding.
I currently have a hdmi cable from lounge amp to kitchen soundbar and it happily plays 5.1 in lounge and 2.1 in kitchen simultaneously.
The soundbar is capable of accepting 7.1 input via hdmi.
The baluns are arriving today so I’ll post back later with results.