"I am unsure if the mentioned tool "CRU" it works in this setup."
You should try it; I'm sure it will work for you as well. You can even revert it. Here's somebody explaining how to do it: https://powrusr.com/fix-the-dolby-vision-black-screen-issue-in-games-like-need-for-speed-heat/In a different context, but this is precisely what I did for my second monitor.
Thanks for the URL. I tried it and removed the DV feature from the LG with this tool. Restarted GPU driver. Installed faulty Win11 update (xxx3060). After reboot sadly the green screen was back. Had to uninstall the Win11 update (xxx3060) and pause automatic updates.
But, before uninstalling the update I hooked up the ROG Ally to a 2024 LG OLED. That TV had no trouble displaying the picture with the update installed (xxx3060). Looks like the DV driver from Win11 has some trouble with older DV display devices.