Hi Victor,
thank you very much for taking your time to explain all this. I seriously thought about refunding CC4 after seeing the result and lack of support for Unity. And since I couldn't find any decent hair shader for HDRP online or in the asset store, your tool is a major relief. Don't want to spend my time writing a shader for a paid product promoting Unity support.
Again, Awesome work!I restarted my testing now (fresh new sunday xD). I cloned the git repo and actually followed your installation tutorial this time.
First I've built the HQ + Single Pass Hair materials and loaded the model up in the preview scene. Same result:

Then I created a new empty HDRP Unity project and exported Camilla from scratch using the "CC4 Camilla Facial Demo" project example and did the following:
1.) Import fbx and textures
2.) Open Importer, select model and set it to Single Pass Hair
3.) Open Preview Scene
4.) Click Build Materials
5.) Weird Hair result
Here's a short video on the whole process (Single-Pass, then Two Pass). Maybe I am doing something wrong. If so just forget about it, will use the other shader anyway, since my project is HDRP and not URP:
I will test it with the CC4 LOD feature and also try the built in materials and test with them, because I want to check out the tesselation feature for LODs of my characters or Game Base (thus why I asked for the ability to produce Built-In Shaders and Two Pass Hair at the same time, even though the hair will then not have tesselation).