Any ideas how to broadcast TV audio to the LAN using RPi?

Most programs that do this introduce significant latency. Delays of several seconds is not uncommon. If you need sync with the display you’re out of luck.
I am sure it is possible though. In the Pro (studio, live) world latency is enemy number one. Whether there is hardware/software for RPi in the Pro sphere is a different question.
Roon is working on an ‘input module’ where audio from ‘something’ can be input to Roon and distributed normally from there. My understanding is that this is low priority.
There is a thread here on the subject: