Assessing climate policies that involve temporary overshoot of temperature targets requires an accurate representation of carbon cycle and climate dynamics. Here, we compare temperature overshoot climate policies obtained with the dynamic integrated climate–economy (DICE) integrated assessment model using two different climate-carbon cycle sub-models: first, the original DICE implementation, and second an implementation of the finite amplitude impulse response (FaIR) simple climate model. We analyze in a cost-effectiveness framework the minimum abatement and carbon dioxide removal costs for compliance against a (future) ceiling on temperatures. In our setup, the magnitude of the overshoot is not limited by temperature impacts, but simply by the temperature dynamics such that from a certain compliance date onwards a temperature ceiling cannot be exceeded anymore. We show that the rather sluggish temperature response and underestimation of carbon sinks in the most recent version of DICE implies that the additional future temperature change after a cessation of a given CO2 emission scenario is significantly overestimated compared to the zero emission commitments obtained with FaIR and complex earth system models. However, investigating climate policies which allow for a temporary temperature overshoot, this inertia translates into more than twice as high optimal carbon prices compared to FaIR and consequently in rather strict climate policies. For compliance with the 1.5 °C target from 2100 onward and non-CO2-warming of 0.2 °C, the mean optimal carbon prices in the year 2030 are 173USD/tCO2 and 56USD/tCO2 for DICE and FaIR, respectively. Still, the dynamics towards the target suggest that improved understanding of and accounting for (limited) reversibility of vulnerable Earth system components is required to derive appropriate overshoot climate policies.