The next generation of gravitational wave detectors will improve the detection prospects for gravitational waves from core-collapse supernovae. The complex astrophysics involved in core-collapse supernovae pose a significant challenge to modeling such phenomena. The Supernova Model Evidence Extractor (SMEE) attempts to capture the main features of gravitational wave signals from core-collapse supernovae by using numerical relativity waveforms to create approximate models. These models can then be used to perform Bayesian model selection to determine if the targeted astrophysical feature is present in the gravitational wave signal. In this paper, we extend SMEE's model selection capabilities to include features in the gravitational wave signal that are associated with g-modes and the standing accretion shock instability. For the first time, we test SMEE's performance using simulated data for planned future detectors, such as the Einstein Telescope, Cosmic Explorer, and LIGO Voyager. Further to this, we show how the performance of SMEE is improved by creating models from the spectrograms of supernova waveforms instead of their timeseries waveforms that contain stochastic features. In third generation detector configurations, we find that about 50% of neutrino-driven simulations were detectable at 100 kpc, and 10% at 275 kpc. The explosion mechanism was correctly determined for all detected signals.

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