Which floor plan software for a complicated house?

That already looks good :) I will take a close look at the software first.

The different levels are not primarily important to map now - it would be more important to be able to include the walls of the ground floor in the planning of the upper floor.

But actually any house design program can do that Ground floor with lines from the upper floor Upper floor with lines from the ground floor:

Generally speaking, we need a plan that at a glance shows exactly what is below/above a certain room.

Exactly: you need a plan based on existing measurements. You would first have to put those into a sketch yourself.
You are not alone in that; someone else from the family who knows how to do it would have to make the sketch.

The measurements would have to be entered into a program. You would do that with the help of this sketch. You would spend too much time on that.
That makes a (expensive) program unnecessary.
It's sometimes already a hassle to draw without guidelines, i.e. without existing conditions, but rather an imagined new building.

What you need is a section with levels for the stairs because of the different heights.
For the exact position of the bathtub you need a tape measure and the installation dimensions of the bathtub itself.
The program won't help you with that.
You can do the free room with Homebyme.

Cadvilla should also include the VA Hausdesigner ;)

The professional version is shown to me as cheaper...

The goal in the medium term is to be able to determine the position of the new staircase. [...] Generally speaking, we need a plan that at a glance provides information about exactly what is below/above a certain room.

If that is the problem description, I don’t think the plan drawing software — aside from the already mentioned drawback that those with high potential also require a lot of familiarization — is the solution (and not even the direction where the solution would lie).

The representation here is not demanding, so a drawing program, even from the shareware section, would be suitable; rather, the measurement is the "level" where the problem lies.

Exactly where the work is tricky here, planning software does not relieve you of the work at all.

I would therefore recommend the following approach: first you need a basic sketch, gladly hand-scribbled, at a rough scale. You make one for each part of the building and for each level. In the first step, we’ll simply call the levels floors.

Then you select suitable points whose vertical alignment is certain: for example, an existing stairwell with the floor-ceiling "crossing" wall adjacent to the staircase.

You then look for a reference point in the ground plan of the ground floor, which you can track here by the floor-spanning line of sight into the upper floor. From this point on the plan, you then create a point in space by assigning it a height, for example 1.00 m above the top edge of the finished floor.

You now place a spirit level with a protractor against the aforementioned wall, because you want to ascertain whether the wall goes straight up (or how much it deviates from vertical), since you need a comparable reference point on the upper floor (and to find out where its plumb line is relative to the ground floor reference point).

On both floors, you then look for further reference lines whose relative positions to the "zero point" you can determine. Such lines run through door openings, so they can be tracked from the corridor into the rooms. Confirmed right angles (e.g. in the form of floor tiles) are your friend.

Once the principle is understood, visualization is just the cherry on top and can be done with ordinary off-the-shelf software.

For the next chapter (relating the building parts to each other), a transparent water-filled hose helps to align the height measurement points.