Financing single-family house in southern Lower Saxony - SPK on strike!

then you probably didn’t have a good mortgage broker. It’s good that the local savings bank handled it for you. That it doesn’t always have to be that way can be seen from the OP’s story. Especially with savings banks and cooperative banks, you can’t generalize from one to all others (statistically it’s not very meaningful anyway, but even less so here), since they operate independently and only cooperate, but in principle each can set its own criteria.
I would rather look for an independent mortgage broker. Interhyp is not independent; as someone already said above, they belong to ING.
There are also fee-based advisors who provide truly independent advice (no commission). Whether that is absolutely necessary should be decided individually.

Are there any recommendations for good FB?

Is it advisable to do something like this exemplarily through Dr. Klein or similar?

Definitely useful for comparison. We once outlined our framework at Dr. Klein as well, and the advisor also showed us the screen with the banks and which ones would finance us under which conditions. If something was marked in red, he quickly knew which information was missing.
Whether it has to be Dr. Klein is another question. There are also smaller ones. As so often, it depends on the individual. I currently have no comparison either, as it is not concrete enough yet.

I would rather look for an independent mortgage broker. Interhyp is not independent; as someone already mentioned above, they belong to ING.

However, you cannot really say that in general terms. Although Interhyp is a subsidiary of ING, in principle, its advice and brokerage should be regarded as independent. This means that they by no means only broker to ING, as this might be the impression here. And ultimately, the individual brokers also live off their commissions, so they are of course interested in offering the customer a good and attractive solution that works – independently of ING.

I did not claim that they give everything to ING (which is not possible anyway, since ING does not want everything). I only said that Interhyp is not independent (which is true) and that I would prefer an independent one. Of course, there is an implication that I believe (not know) that they tend to act according to guidelines rather than in the interest of the customer when in doubt. And I admit that I do not know this for sure, let alone could prove it. How do you know the internal processes there so well?

You will certainly find a solution. Interhyp, Dr. Klein, etc. can be helpful. However, I also throw in "fairhypo" from your region (Braunschweig). If the senior (Thomas) seems somewhat "peculiar" to you, that should not deter you :) He is a very open guy, experienced, dedicated, and always easy to reach. Maybe you want to give it a try with him. PS: He also has somewhat of a reputation for finding solutions where others have long given up ;)