The integration of a wireless backhaul network between a macro-cell base station (MBS) and multiple small-cell base stations (SBSs) with full-duplex (FD) technology at the SBSs offers significant potential to enhance spectrum efficiency in millimeter-wave (mmWave) communication systems. However, the FD mode also introduces an inherent trade-off between better time utilization and increased network interference, necessitating careful optimization of duplex modes at each SBS. In this paper, we propose a novel framework that jointly optimizes both the duplex mode selection for each SBS and the design of hybrid beamforming (BF) at the MBS and SBSs to maximize network throughput. Our framework alternately tackles two key subproblems: hybrid BF design and SBS duplex mode selection. For hybrid BF design, we employ a two-stage approach that combines a codebook-based radio frequency beam selection via a greedy algorithm with baseband BF optimization through a generalized power iteration method. For SBS duplex mode selection, we introduce the worst-mode switching (WMS) algorithm to efficiently find a near-optimal solution in polynomial time. Simulation results demonstrate that the proposed joint algorithm not only has low computational complexity but is also robust to various interference scenarios, including different ADC resolutions and varying levels of residual self-interference suppression. This adaptability results in superior sum-rate performance compared to conventional hybrid BF schemes and configurations where all SBSs operate exclusively in either full-duplex or half-duplex mode.