Core-collapse supernovae are among the most promising yet still undetected sources of gravitational waves. A future detection would provide a direct view of the physical processes occurring deep inside a collapsing star. In this work, we investigate how networks of current and future gravitational-wave detectors can constrain the properties of rapidly rotating core-collapse supernovae using their characteristic core-bounce and early post-bounce signals. Using deep-learning techniques, we estimate the peak frequency, rotation rate, and signal amplitude from noisy detector data and compare the performance of different detector-network configurations. We find that detector networks improve both parameter recovery and sky coverage. For current-generation networks, estimation of the peak frequency is possible out to about 15 kpc, while the rotation rate remains recoverable out to distances exceeding 100 kpc. Third-generation observatories extend these distances by approximately a factor of three to four.