I know nothing about Chromecast other than what I see on the web page for it so I can’t really offer real help. My guess is you are connecting the Chromecast to a HDMI port on the monitor so any signal from the Chromecast, including audio, goes over HDMI. I don’t know the specifics of the monitor but I doubt that it can send audio from an HDMI input back over DisplayPort to the dock but that is a guess. Does the LG monitor have an audio output port (analog line-level)?
You might be able to find a device with an HDMI input that can pull the audio stream and send it out via optical or line output level (might be called an HDMI audio extractor). You might be able to put that into either the line or optical input of the TV adapter and the laptop audio stream into the other adapter input. Or look into an optical switch to select which TosLink input goes to the TV adapter (laptop via dock optical port or HDMI device that puts out audio via optical port).
The above is totally guesswork but there are probably additions to your already complicated setup that might allow what you want.
Is the TSPlus a USB or Thunderbolt dock? I have a TS3Plus that uses Thunderbolt 3 and doesn’t work with a USB connection to the computer - I use a Thunderbolt 3 active cable to connect to an iMac. The iMac audio tools allow selection of both input and output for audio. With my setup, I can see an input to the Mac from the Caldigit Thunderbolt 3 audio but I don’t have anything connected to that. From the Caldigit docs, it looks like the only audio input to the TS3Plus is a line level analog. I have a Dell UP2718Q monitor connected to the TS3Plus - for audio output one item in the long list is the Dell monitor as it supports audio over DisplayPort but the audio output from the monitor is only analog line output. That can be used to drive a sound bar and can output the sound from either the DP or HDMI output. If you monitor has a similar audio output port you could put that into the TV adapter analog stereo input.