What I have experienced with Ethernet to HDMI was in conference room. Went through several of them before we found a stable one to work.
In the end the one that is currently being used (2 years now) does not have as bright of a picture as hooking it up direct through a HDMI cable.
I know that your question is about audio however in the end, the results although it did work was degraded. I have no way of knowing exactly what is being degraded. Is it just the video and or also the audio.
Because of that even though I cannot categorically say for sure what all is degraded, I would use the HDMI cable.
But that’s just me.
–MD