For 20 bucks you can get an hdmi audio extractor[1]. So for a little bit extra money you can simply connect the regular chromecast to your amplifier.
[1] https://www.amazon.com/slp/hdmi-audio-extractor/xc32yz93wutk...
[1] https://www.amazon.com/slp/hdmi-audio-extractor/xc32yz93wutk...
And I can't find anything about whether it will resample the audio (dealbreaker), or how it will negotiate that.
I'm sure there will be a couple of unforseen issues as well. My chromecast occasionally disconnects itself, which means that you would have to rewire the hdmi cable to the TV just to have a clue of what is going on (can my parents or house gets figure that out when I'm not around?).
And yet another device and power brick.