So, after being so well advised here, I of course want to inform you how the offer ultimately looks that I received yesterday.
As was suspected several times, the offer was ultimately associated with compromises.
We received several recommended plots for calculation that were around 50,000 euros. The house would come to 182,000 euros without a basement, ready to paint, and 136 sqm. Additionally, 20,000 euros were calculated as additional construction costs, which, in my opinion, is very tight (authorities such as notary, property taxes, etc. 3,500, house connections 6,000, civil engineering costs 8,000 (but this was calculated with a basement; without basement excavation 5,000), buffer 2,500).
With a basement, the house would be 20,000 euros more expensive and would be outside the budget, as some here already suspected. There is also the uncertainty whether the additional costs will be sufficient, because some things on the construction site are still the responsibility of the builders. It's all a shaky calculation.
We will now get offers from two other regional providers in Saarland (however, solid construction) and then decide whether this makes sense or not, and whether we continue to look for existing properties.