How large is the property, just out of interest?
I would be a bit cautious with general contractors like Allkauf Haus. I could be mistaken, but I once watched quite a few construction defect documentaries on YT and recall that they showed up there quite often. Of course, this can vary regionally, but in general, the construction price for the Stuttgart area also seems very cheap to me.
For you, purely from a household service perspective, it's really a tight squeeze. With that loan amount, we would prefer not to have a 5 in front based on gut feeling (still undecided), but we have significantly higher incomes.
If I were you, I would fire up interest calculators, there are good ones online. With very high loan amounts, you also quickly run into the problem that the loans would run very long at 2% amortization. Well, you are still quite a bit younger than us, but the interest rate uncertainty then comes on top for a follow-up financing with presumably high remaining debt.
Because of the interest rate development this year, I am also a bit worried, but it cannot be predicted exactly how things will continue. I have also read sound arguments suggesting that the development could relax somewhat. Still, a general upward trend can be seen, even before the war.
Would it be important for you to have the house paid off around retirement? You could try calculating different interest scenarios and see how long you would need.