Thank you for your answers!!
And finally: The two-flue chimney (because of controlled residential ventilation) costs about 5000€. The pressure monitor about 2500€. Then another 5000-7500€ for the stove and the control system... Plus chimney sweep every year.
Thank you for your opinion. Pro and con opinions are slowly balancing out! Some are completely sure about it.... it brings quite a bit with an air heat pump, and others are completely convinced of the opposite.
Regarding the costs: We will definitely have a chimney. It is only a question of whether the chimney insert will have water guidance or not. I am leaving out the two-flue chimney from the cost calculation, since it is already installed because of controlled residential ventilation. The pressure monitor is said to cost about 600-700 €, which I also need because of the controlled residential ventilation. My heating engineer will prepare everything needed up to the chimney for 400 €. In addition, there are the costs for an 850-liter storage tank instead of 300 liters. The 850-liter storage tank is divided so that the hot water is only buffered in one part of the tank. The chimney insert will cost me more, and the control system connection will be handled by the heating engineer, but the electrician and chimney sweep must also be involved.
In total, I have additional costs of about 3,000-4,000 € for the water-guided chimney compared to the chimney without water guidance. And I find it difficult to decide whether this 3-4k investment will pay off in the near future, since some are completely convinced and others are not.
It should not get too warm, as the underfloor heating is regulated with the water-side heat from the chimney and the air heat pump, and the excess heat does not flow into the underfloor heating (my assumption).
I hope I was able to fairly well present my difficulty in deciding.
Best regards Alex