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Upcoming wide-field surveys will discover thousands of new strongly lensed quasars which will be monitored with unprecedented cadence by the Legacy Survey of Space and Time (LSST). Many of these quasars will undergo caustic-crossing events over the 10-yr LSST survey, during which the quasar's inner accretion disc crosses a caustic feature produced by an ensemble of microlenses. Such caustic-crossing events offer the unique opportunity to probe the vicinity of the central supermassive black hole, especially when combined with high cadence, multi-instrument follow-up triggered by LSST monitoring. To simulate the high-cadence optical monitoring of caustic-crossing events, we use relativistic accretion disc models which leads to strong asymmetric features. We develop analysis methods to measure the innermost stable circular orbit (ISCO) crossing time of isolated caustic-crossing events and benchmark their performance on our simulations. We also use our simulations to train a convolutional neural network (CNN) to infer the black hole mass, inclination angle, and impact angle directly from these light curves. As a pilot application of our methods, we used archival caustic-crossings of QSO 2237+0305 to estimate the black hole mass and inclination angle. From these data, two of these methods called the second derivative and wavelet methods measure an ISCO crossing time of 48.5 and 49.5 d, corresponding to a Kerr black hole mass of M BH = (1.5 ± 1.2) × 109 and (1.5 ± 1.3) × 109  M⊙, respectively. The CNN inferred log10(M BH/M⊙) = 8.35 ± 0.30 when trained on Schwarzschild black hole simulations, and a moderate inclination of i  = 45 ± 23°. These measurements are found to be consistent with previous estimates. [ABSTRACT FROM AUTHOR]