In Escherichia coli the response to DNA damage shows strong cell-to-cell-heterogenity. This results in a random delay in cell division and asymmetrical binary fission of single cells, which can compromise the size homeostasis of the population. To quantify the effect of the heterogeneous response to genotoxic stress (called SOS response in E. coli) on the growth of the bacterial population, we propose a flexible time-continuous parametric model of individual-based population dynamics. We construct a stochastic model based on the "adder" size-control mechanism, extended to incorporate the dynamics of the SOS response and its effect on cell division. The model is fitted to individual lineage data obtained in a 'mother machine' microfluidic device. We show that the heterogeneity of the SOS response can bias the observed division rate. In particular, we show that the adder division rate is decreased by SOS induction and that this perturbative effect is stronger in fast-growing conditions.