Our current understanding of the core-collapse supernova explosion mechanism is incomplete, with multiple viable models for how the initial shock wave might be energized enough to lead to a successful explosion. Detection of a gravitational-wave signal emitted in the initial few seconds after stellar core-collapse would provide unique and crucial insight into this process. With the Advanced LIGO and Advanced Virgo detectors expected to approach their design sensitivities soon, we could potentially detect this signal from a supernova within our galaxy. In anticipation of such a scenario, we study how well the BayesWave algorithm can recover the gravitational-wave signal from core-collapse supernova models in simulated advanced detector noise, and optimize its ability to accurately reconstruct the signal waveforms. We find that BayesWave can confidently reconstruct the signal from a range of supernova explosion models in Advanced LIGO-Virgo for network signal-to-noise ratios ≳30 , reaching maximum reconstruction accuracies of ∼90 % at SNR ∼100 . For low SNR signals that are not confidently recovered, our optimization efforts result in gains in reconstruction accuracy of up to 20%-40%, with typical gains of ∼10 %.