Finally the Peltier system is up and running. The solution with the L298N works quite well.. Here some images of temperature curves we logged with it... (it still needs some fine tuning, but it works!)

The images - Room temp to 15 °C. Cycles of 35 to 25 °C and room temp to 35 °C

Here after some more tuning:

5 min periods for each temp. 5min for settling into room temp (after 35°C) than switching between 35°C and 15°C (4X) and one period at 19°C and more 5min with peltier off (the bump comes from heat dissipating from the bottom side of the peltier to the top side, where temperature is measured).