Depending on how and where the staircase is anchored, I would make a formwork and drill a few holes sideways into the existing concrete ceiling. Insert a few steel rods there to connect the new concrete. Then pour a screed concrete from above and let it dry. If the roof above is a cold roof, do not forget the insulation.
That would also be my preferred idea. Since it is a cold roof, I am considering buying a new (matching) staircase with a better U-value.
Although you say you don’t want much storage, is there really enough other storage space available?
Standing height at the highest point is about 1 meter. We are building with a basement, so there is enough storage space there.
Whether this is really just about backing the frame on one side (and which one), is a question of the staircase’s construction. [...] I consider clarifying that alone more expensive than the bargain potential when buying the staircase.
I would indeed have to take another close look at the staircase. It is already at my home and was basically free.
The unsuitable dimensions and the mediocre U-values make me currently tend to buy a matching staircase after all. Thanks for your opinions!