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Streamflow elasticity (εₚ) is vital for assessing future water resources, but in human-managed catchments, existing estimation methods often overestimate εₚ due to neglected anthropogenic controls like dam operations or groundwater withdrawal. We address this gap by quantifying εₚ across 78 highly regulated data-scarce catchments in Peninsular India, explicitly accounting for the effects of large-scale storage changes and unobserved water withdrawals. We compare a traditional linear regression model (LM) against two proposed probabilistic frameworks: a Bayesian regression model (BM) and a Bayesian hierarchical model (BHM), across long-term daily average, monsoon sum, and annual sum temporal aggregations. The LM yielded physically implausible elasticity estimates (up to 4.8), with the unexplained variance erroneously attributed to precipitation. Bayesian models that isolates non-climatic factors, achieved substantially improved model skill and εₚ values that were substantially lower than LM-based estimates and more physically consistent, values ranging from 0.28 to 0.68. These revised estimates confirm that the true streamflow elasticity of these basins is far less sensitive than inferred by baseline models. Both Bayesian models (BM and BHM) yielded nearly identical εₚ estimates, and validation checks like comparison of storage change estimates with aridity index showed consistent trends confirming the robustness of the proposed methodology. Elasticity was found to be highest for annual sum aggregation (up to 0.68), reflecting the amplified annual-scale sensitivity of streamflow to precipitation. This work provides a reliable framework for separating climatic signals from anthropogenic noise in regulated systems. The resulting lower εₚ values are crucial for generating realistic climate change impact projections.