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

If you have doubts that the increase is lawful, you can probably have it legally reviewed. The initial consultation usually doesn’t cost much(?) and there might already be some findings.

What you then do with the information gained is the next question. If the answer is no, they shouldn’t have done it that way, then the question is whether blocking 100% will really get you anywhere. At least it would give you a stronger negotiating position, however you then use it (there’s quite a bit of room between swallowing every increase completely and blocking everything entirely).
And if the answer is yes, yes, they were allowed to do it that way, it would also be important for you to know whether they may do so in the future (and if so, to what extent) and what your options are. So, I have no legal knowledge, but if my bank changes the terms for my checking account, I can usually terminate with shorter notice, etc., or object, but that can also have consequences. Something like that.

So for me, it would first be important
to know if what has happened so far was correct and what else might come your way. And probably only someone who knows construction law and dissects your contract again can really explain that to you. (Just because you get advice doesn’t mean you immediately have to sue or anything)

the one on the plan is not a prefabricated garage.

No, you misunderstand. You signed the offer before the appointment (03.03.). That means acceptance of the offer is equivalent to a contract. Therefore, the contractor cannot adjust the price. Unless they are threatened by insolvency.

Only if you have conducted another negotiation would that speak against an existing contract.

If the contractor wants to change the price, you are off the hook from the contract. Generally, you should have made little advance payment so far and thus the damage you have to compensate the contractor for is rather low.

This is just my opinion. The exact procedure is specified by a lawyer. They know all the details.

I would also like to see if a redesign for UG and EG makes sense.

With a story height of 2.60 in the basement, the minimum height (2.40m) of living spaces is hardly achievable. I would like to see the floor plan. Is there a plan of the property with height indications?

I looked at the plans again. When entering the hallway on the ground floor, you walk directly towards the stairs at a distance of 1 m, who plans something like that?

Sorry, I totally forgot that here. A small granny flat is planned in the basement, we are getting 2x the funding for KfW 55. It was already approved for us in November 2021.

And the mini granny flat is never supposed to be rented out but is only for the teenage daughter. Since we had to make a basement anyway, we just planned it to receive the funding.