Required build height is 12mm, but the desired vinyl has only 4.5mm thickness

You have to face one death. Either you have problems now (that's why you're posting here) or later on.

Options:
1. Use leveling compound, then the vinyl fits now, but no other covering later (or removal of the leveling compound)
2. Live with the small step
3. Choose a different floor covering. With 12mm, tiles, glued two-layer parquet, or "thicker" design flooring on HDF with cork insulation or laminate are possible

However, the plastic floor is simply too expensive for me compared to, for example, parquet. Not to mention optics and feel.

OK to answer the questions:

Have you ever chosen floor coverings with your wife/girlfriend (or with both at the same time )? I think that answers the question

As already said, immense effort, so that in the end the raw floors of this floor no longer lie on one level and you are forever at 4? No.
The option 3 mentioned by nordanney is probably the most acceptable variant.

Why did I even ask this question here in the forum? Well, I had hoped that someone would say "yes, your problem is a common question, there is the following product xyz with which you can arbitrarily increase the build-up height". Apparently not, but the thread still helped me. =)