Exposure to environmental stressors, including certain antibiotics, induces stress responses in bacteria. Some of these responses increase mutagenesis and, this way, potentially accelerate resistance evolution. Many studies report increased mutation rates under stress, often using the standard lab approach of fluctuation assays to estimate the mutation rates. However, many stress responses are heterogeneously expressed in bacterial populations, which existing estimation methods have not yet addressed. We develop a population dynamic model that considers heterogeneous stress responses (subpopulations of cells with the response 'off' or 'on') and derive the mutant count distribution arising in fluctuation assays under this model. We then implement a computational method to estimate the mutation-rate increase specifically associated with the induction of the stress response. Using simulated mutant count data, we show that our inference method allows for accurate and precise estimation of the mutation-rate increase, provided induction of the response also reduces the division rate, as is the case for the SOS response. Interestingly, we find that existing methods assuming a homogeneous stress response can also accurately infer the increase in population mean mutation rate. Without additional experiments, it is not generally possible to distinguish between a model with heterogeneously expressed stress responses and a model in which mutants have a differential fitness compared to non-mutants.