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We apply common gravitational wave inference procedures on binary black hole merger waveforms beyond general relativity. We consider dynamical Chern-Simons gravity, a modified theory of gravity with origins in string theory and loop quantum gravity. This theory introduces an additional parameter $\ell$, corresponding to the length-scale below which beyond-general-relativity effects become important. We simulate data based on numerical relativity waveforms produced under an approximation to this theory, which differ from those of general relativity in the strongly nonlinear merger regime. We consider a system with parameters similar to GW150914 with different values of $\ell$ and signal-to-noise ratios. We perform two analyses of the simulated data. The first is a template-based analysis that uses waveforms derived under general relativity and allows us to identify degeneracies between the two waveform morphologies. The second is a morphology-independent analysis based on BayesWave that does not assume that the signal is consistent with general relativity. The BayesWave analysis faithfully reconstructs the simulated signals. However, waveform models derived under general relativity are unable to fully mimic the simulated modified-gravity signals and such a deviation would be identifiable with existing inference tools. Depending on the magnitude of the deviation, we find that the templated analysis can under perform the morphology-independent analysis in fully recovering simulated beyond-GR waveforms even for achievable signal-to-noise ratios $\gtrsim 20{-}30$.