To not leave this as an open question here and in case someone has a similar problem later:
A technician was here and looked at the system. A control valve is defective, so the system cannot regulate itself. This causes the internal pressure to be twice as high as normal despite reduced performance, which then triggers the high-pressure fault. The error has nothing to do with cooling and hot water; the defect was probably there from the beginning but has only now become noticeable through faults due to higher brine temperatures.
The temperature differences between supply and return in hot water preparation have also been clarified: The heat pump only has one temperature sensor for the return flow, and it is located in the heating circuit, so it is not integrated into the circuit during hot water preparation and therefore shows nothing useful.
The control valve will be replaced soon, then we will see further. However, it will be a major operation, installation time about 4 hours.