Remember those HDCP rules.
That pass through might be restriicted, blocked or degraded through the HDTV. Bluetooth is usually stereo (99.99% of the time) so if you are trying to pass 5.1 sound it might go silent.
This area is well worn but I understand folk are going to bump into this over and over.
Also, an input has no sound output. I'm sure you meant something else but wanted to be painful crystal clear because I did encounter too many setups where they connected the TV's input jacks to their receiver's input jacks and wondered why nothing happened.