Inverse design has been widely studied as an efficient method to reduce footprint and improve performance for integrated silicon photonic (SiP) devices. In this study, we have used inverse design to develop a series of ultra-compact dual-band wavelength demultiplexing power splitters (WDPSs) that can simultaneously perform both wavelength demultiplexing and 1:1 optical power splitting. These WDPSs could facilitate the potential coexistence of dual-band passive optical networks (PONs). The design is performed on a standard silicon-on-insulator (SOI) platform using, what we believe to be, a novel two-step direct binary search (TS-DBS) method and the impact of different hyperparameters related to the physical structure and the optimization algorithm is analyzed in detail. Our inverse-designed WDPS with a minimum feature size of 130 nm achieves a 12.77-times reduction in footprint and a slight increase in performance compared with the forward-designed WDPS. We utilize the optimal combination of hyperparameters to design another WDPS with a minimum feature size reduced to 65 nm, which achieves ultra-low insertion losses of 0.36 dB and 0.37 dB and crosstalk values of -19.91 dB and -17.02 dB at wavelength channels of 1310 nm and 1550 nm, respectively. To the best of our knowledge, the hyperparameters of optimization-based inverse design are systematically discussed for the first time. Our work demonstrates that appropriate setting of hyperparameters greatly improves device performance, throwing light on the manipulation of hyperparameters for future inverse design.