KFW40EE house - fireplace fire but it should not get too hot

With real green electricity or photovoltaic power from the roof, about ~ factor 10 less again.

Yeah. A little bit for sure.
Numbers: One kilo of wood brings about one heat-kWh into your house. In doing so, 1.8 kg of CO2 is produced. (sounds crazy, but that’s how it is ;-)
Producing one heat-kWh via the heat pump emits 0.14 kg of CO2. With genuine green electricity or photovoltaic power from the roof, about ~ ten times less again.

So wood versus heat pump is somewhere between 10:1 and 100:1.
In other words: This "just a little cozy heat on the weekend!" doubles your heating system’s CO2 emissions.

if it rots unused in the forest.

Yeah. Probably a bit.
Numbers: One kilo of wood brings about one heat-kWh into your house. This results in 1.8 kg of CO2. (sounds crazy, but it's true ;-)
Producing one heat-kWh with a heat pump emits 0.14 kg CO2. With real green electricity or photovoltaic power from the roof, about a factor of 10 less again.

So wood to heat pump is somewhere between 10:1 and 100:1.
In other words: This "Only a bit of cozy warmth on the weekend!" doubles the CO2 emissions of your heating system.