I owe everyone an apology, especially @FordGuy61 for so much work testing. His results made me try more and I played with Video format settings in my Chromecast. I was getting weird results and some video artifacts, so I decided to Restore factory settings on the Chromecast (I have changed the TV set recently).
After setting it up again (Dolby Vision Low latency 4k 60FPS, Match format to content Seamless), I get correct TV picture mode depending on the content. And all audio transcodes correctly, no more infinite spinners.
Let‘s consider this topic closed. My WebOS Plex app still has problems with this soundbar, but I will open a separate topic for that. Chromecast is finally usable as my audio delay problems are gone on the new TV set (LG C2). It does stutter impossibly on some rare content and I can’t adjust huge subtitles, but I may now actually consider using it instead of WebOS app.
Thanks everyone for your help!