Hybrid beamforming (HBF) multi-user multiple-input multiple-output (MU-MIMO) is a key technology for unlocking the directional millimeter-wave (mm-wave) nature for spatial multiplexing beyond current codebook-based 5G-NR networks. In order to suppress co-scheduled users' interference, HBF MU-MIMO is predicated on having sufficient radio frequency chains and accurate channel state information (CSI), which can otherwise lead to performance losses due to imperfect interference cancellation. In this work, we propose IABA, a 5G-NR standard-compliant beam pair link (BPL) allocation scheme for mitigating spatial interference in practical HBF MU-MIMO networks. IABA solves the network sum throughput optimization via either a distributed or a centralized BPL allocation using dedicated CSI reference signals for candidate BPL monitoring. We present a comprehensive study of practical multi-cell mm-wave networks and demonstrate that HBF MU-MIMO without interference-aware BPL allocation experiences strong residual interference which limits the achievable network performance. Our results show that IABA offers significant performance gains over the default interference-agnostic 5G-NR BPL allocation, and even allows HBF MU-MIMO to outperform the fully digital MU-MIMO baseline, by facilitating allocation of secondary BPLs other than the strongest BPL found during initial access. We further demonstrate the scalability of IABA with increased gNB antennas and densification for beyond-5G mm-wave networks.