There is a palpable shift in mainstream attitude towards geoengineering, seen now as a potential part of a climate policy mix. Still, no-one wants to get on a slippery slope, compounding the risks, and, therefore, we should ask ourselves what is the minimal geoengineering that we can get away with. Such questions lead mathematically to inverse problems. Solving them is feasible only with lightweight models of the climate system, various types of which are nowadays often referred to as emulators – some more accurate than others. Here we develop an emulator using nonlinear response theory and apply it to two paradigmatic inverse problems relevant to climate policy. First, we investigate the attainability of the coveted Paris15 temperature targets. Second, through a simple multi-stable model, we determine what it takes to save the Greenland ice sheet (GrIS) as we know it. Our results suggest, first, that as things stand presently, solar radiation management geoengineering will likely have to be part of our climate policy mix. Second, we demonstrate also the importance of precisely knowing not only the stable but also the unstable so-called Melancholia states of climate tipping elements, such as the GrIS, as miscalculations can lead to acting too late.