We present a method to efficiently approximate the response of atmospheric-methane mole fraction and δ13C–CH4 to changes in uncertain emission and loss parameters in a three-dimensional global chemical transport model. Our approach, based on Gaussian process emulation, allows relationships between inputs and outputs in the model to be efficiently explored. The presented emulator successfully reproduces the chemical transport model output with a root-mean-square error of 1.0 ppb and 0.05 ‰ for hemispheric-methane mole fraction and δ13C–CH4, respectively, for 28 uncertain model inputs. The method is shown to outperform multiple linear regression because it captures non-linear relationships between inputs and outputs as well as the interaction between model input parameters. The emulator was used to determine how sensitive methane mole fraction and δ13C–CH4 are to the major source and sink components of the atmospheric budget given current estimates of their uncertainty. We find that our current knowledge of the methane budget, as inferred through hemispheric mole fraction observations, is limited primarily by uncertainty in the global mean hydroxyl radical concentration and freshwater emissions. Our work quantitatively determines the added value of measurements of δ13C–CH4, which are sensitive to some uncertain parameters to which mole fraction observations on their own are not. However, we demonstrate the critical importance of constraining isotopic initial conditions and isotopic source signatures, small uncertainties in which strongly influence long-term δ13C–CH4 trends because of the long timescales over which transient perturbations propagate through the atmosphere. Our results also demonstrate that the magnitude and trend of methane mole fraction and δ13C–CH4 can be strongly influenced by the combined uncertainty in more minor components of the atmospheric budget, which are often fixed and assumed to be well-known in inverse-modelling studies (e.g. emissions from termites, hydrates, and oceans). Overall, our work provides an overview of the sensitivity of atmospheric observations to budget uncertainties and outlines a method which could be employed to account for these uncertainties in future inverse-modelling systems.