What is the right course of action here?

You answered correctly. In this situation, you need to brake carefully because the road is snowy and slippery, and you must allow the pedestrian on the right to cross since there is a marked pedestrian crossing. In Germany, you are required to give way to pedestrians at marked crossings, even if road markings are obscured by snow. Careful braking is also necessary to avoid skidding on the icy surface. So, both actions—braking carefully and letting the pedestrian cross—are the right response.