Unpredictability of construction costs - Will the project fail because of it?

Definitely postpone the garage in terms of timing and use the budget as a buffer. It costs at least 20k, right? You have a large basement, so you don't (initially) need a garage.

And talk to the construction company about what options they see. For example, removing floor coverings and doing it yourself. Reducing the outdoor area to the essentials (just gravel instead of paving).

Do you already have a child? If not, postpone the attic conversion including the bathroom and use the granny flat as a bedroom in the meantime.
Someone already mentioned buying and storing materials.

Be flexible overall! Your total financial situation is unknown, but if it seems feasible in any way and you have a construction company as a partner you can talk to, I would go for it.

Exactly, the garage costs about 20000 but can still be cut. The first offer still included a ventilation system for 20000, which has also been cut + a planned buffer of 20000.

So as of now, we would have had a 60000 buffer and then the construction company came with the price increase. That was already after the loan approval.

The child is already here, so we should already finish the upstairs.

I would first look at what is stated in the KFW funding conditions regarding the granny flat.

It is not really that important in the first place, because the grant can be claimed up to 2.5 years after approval. By then, you should be able to complete the remaining work on your own.

I have been wondering the same thing the whole time. Of course, it is difficult to judge from a distance without knowing your correspondence. But normally, either both parties have a contractual obligation, or none at all. Unless you signed something stating that the prices are non-binding or similar. Have you ever posted the wording somewhere? I am also wondering what legal binding a "signed offer" has? Normally, you conclude a work contract, which is binding.

That's exactly it. We can't get out, but they can do whatever they want, that can't be right? The following wording was in it: "The contract stated that the prices are binding until 03.03., for a later order placement they reserve the right to adjust prices." However, the order placement was on 14.02. And there was nothing about non-binding or similar either.

Even if you remove the garage now, you still have to create parking spaces. How complicated will this be for you because of the slope? You also need additional parking spaces because of the [EWL].

Where can you actually get a garage for 20k€?

But the offer did not yet include fixed prices for the other trades except for the shell construction etc.? Then what were the fixed prices for at all if the shell construction is now becoming more expensive? On what basis was the price increased? Roughly speaking?

As you already say, it makes no sense if you are bound to a contract that only binds you but allows the contractor price adjustments in seemingly unlimited amounts. Such a contract would be every builder's dream.
Get a lawyer for construction law or contract law and go through it with them.
However, avoid if possible sending your builder a letter from a lawyer right away. If it's a small company, they probably act out of legal ignorance, and your relationship will be ruined at first. I know some bricklayers who have their own construction company but do not know the difference between warranty and guarantee.

The company will recoup its additional costs again if you now force them to bear the price increase themselves. Agree on a solution that is acceptable to both sides.