Financing monthly installment €2500 with 40 years term

Why such a meager repayment with that income? €3,000 - €3,500 should easily be possible for you, or are the other expenses perhaps too high? If so, then as a first step I would suggest prioritizing.

The way you have planned it is, in my opinion, absolutely not healthy and carries a pretty high risk.

The fixed interest period is 10 years.

Since we want to keep our monthly payment at around €2500 (concrete offer from VR-Bank Saale-Unstrut: €2546), this results in a term of 40 years.

10 years fixed interest period with a 40-year term does not work for me at all. In 10 years you will almost have the same amount again as follow-up financing, because you have repaid so little. What is the interest rate for that?
I would suggest a combination of a longer fixed interest period and a higher rate, and also check whether you can get a better deal from other banks.

Also, I am skeptical about your calculation. Land + development + house + painter already amount to €637k. With €30k equity, that leaves you just €23k for additional construction costs. Is the land from the construction company or was it brokered by them? Then the land transfer tax alone would probably eat that up. Added to that are utility connections, possibly earthworks + disposal, construction road, kitchen, buffer for upgrades, outdoor facilities. Notary and land registry also like to get their share, a building permit also costs something, etc.

What interest rate does the bank offer for a 10-year fixed interest period? Have you already spoken with other banks?

Since when have you been working in the profession? Of course, it takes a long time to become a tenured teacher, but it does help to estimate your savings/consumption behavior. The low equity makes it very difficult for you with the interest rates, I would definitely choose a longer fixed interest period. Basically, it is of course feasible based on the numbers.

And two times 100% after parental leave is also more difficult than some want to admit.

As a tenured teacher, shouldn't a good interest rate still come out? It certainly doesn't get any safer for the bank.