I got the project migrated to Unity's new Universal Render Pipeline (URP) on 13.3.2021 after multiple days of trying to get it to work. Now, when writing this, it's been so long that I don't even remember what was the issue. Nevertheless, the kaleido effect is now a scriptable renderer feature of the Forward Renderer.

After all the tweaking, I might be able to recreate it myself from a scratch, but this far it's been much easier to just adjust it to my needs. Even though this kind of effect is considered simple, I am really happy I didn't need to think so hard.

Plan B was to ask for help.

Here's also some screenshots of different effects you can achieve with some modifications to the kaleidoscope. I'm hoping to use these for dramatic effect when it makes sense in the storyline. Distortion and warping are easy to achieve, and these are aesthetically pleasing ways to represent chaos, in my opinion.