There is a palpable shift in mainstream attitude towards geoengineering technologies, seen now as potential parts of a climate policy mix. Still, concerning solar radiation management (SRM) in particular, because of the known and unknown undesirable side-effects of various engineering implementations of theirs, it is important to know what is the minimal intervention that can achieve a certain goal. 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 linear and nonlinear response theory and apply it to the minimal SRM problem concerning the Paris 2015 climate agreement, say, with the aim of constraining the global mean surface temperature below a certain limit. Our results suggest that SRM geoengineering, most commonly envisaged as sulfate aerosol injection, will likely have to be part of our climate policy mix, because realistic CO2 abatement effort to come alone cannot restrict global temperatures below the coveted 1.5 ◦C change or below even higher levels of change. Minimal sulfate use for the 1.5 ◦C limit is very likely to dictate immediate and rather abrupt deployment. However, SRM would be no use to achieve such a goal if the geoengineeringfree “asymptotic” temperature is not below the target limit, as it would then need – in the absence of CDR – maintaining SRM “indefinitely”. The latter could be the case even if the temperature response to an anthropogenic CO2 emission pulse is nonmonotonic, and it would be certainly the case if it is monotonic. We show that the model that we use is near the boundary in parameter space between monotonic and nonmonotoninc temperature responses. In the unfortunate case of monotonicity concerning the real Earth system, the only use of SRM would be “buying time” to develop CDR.