Construction delay by the developer - Who bears the rental costs?

There is no legal obligation for the developer to directly avoid the delay (bad weather, etc.), but without any information from him, he is legally in default. That is at least a fact.

What you call a "fact" may be legally correct and is not disputed by me. For an assessment of whether a claim exists or not, the complete information is simply lacking.
The proposal for the letter from is formulated in such a way that I get the impression that there is no experience in business communication or project management. I can address that.

We actually have no idea about contracts in this area and how the basics of business communication look.

So, all in all, it doesn't seem unlikely to me that we may demand something from him. Is he then obliged to provide proof? We were never informed about delays with specific time indications, neither for "extra requests" nor for "bad weather" and also not when craftsmen were sick, errors were corrected, etc.

He is the contractor and we are the clients according to my understanding. At least we had to take out builder's liability insurance and every brick built immediately becomes our property.

To me, this means that we list our previous additional costs and formally demand payment. The question that arises for me is whether we should bring a lawyer on board right away to avoid formal errors or draft a self-written letter.

Thanks in the meantime to everyone who has put thought into it and responded so thoroughly so far!

Is he then obligated to provide proof?

Sure. He also guaranteed a handover/completion date in writing.

We were never informed of delays with concrete time indications, neither with "extra requests" nor with "bad weather" and also not when craftsmen were sick, errors were corrected, etc.

In this respect, I would write to him now that the deadline has been exceeded and that you are incurring costs for accommodation and standby interest.

The question I have is whether we should immediately involve a lawyer to avoid formal errors or write a self-drafted letter.

Good idea. At least for an initial consultation.

The quoted passage from the contract for work is only to establish the delay.

Back then, it also stated what compensation claim we have per day/month.
Maybe read it again carefully.

The quoted passage from the contract for work is only for establishing the delay.

At our company, it also stated what compensation claim we have per day/month.
Maybe read it carefully again.

Regarding this, nothing is stated with us. We still have the point that we pay according to the construction progress in a maximum of x installments and which exactly. 5% is withheld for timely and defect-free completion.

So, all in all, it doesn't seem unlikely to me that we can demand something from him. Is he then obligated to provide proof?

No. You have a date stipulated in the contract, and of course, you can demand that it be adhered to and also that you are compensated for the financial disadvantages that result from it. At first, you can demand anything. You must reckon with resistance to your demand, and it is good to know what this resistance might consist of and what options your service provider has based on the concluded contract. From the comparison of your demand and the expected arguments in case of unwillingness to pay, you can develop a strategy, and that decides how you proceed. The strategy can also be called "lawyer." I like that you rate your experience in such matters as low. That shows that you are not blindly unaware of weaknesses. From your way of speaking, you also give me the sympathetic impression of being a very kind person who prefers to avoid discord. I hardly ever recommend a lawyer, but here I’ll make an exception. My strategy would be to ask the lawyer for advice and help in formulating the demand and to deliver the first letter with the lawyer's wording on your own letterhead securely. If resistance then arises, I would hand over further communication to the lawyer.