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Understanding the computational principles of the brain and translating them into neuromorphic hardware and modern deep learning architectures is critical for advancing neuro-inspired AI (NeuroAI). Here, we develop an experimentally constrained, biophysically realistic model of neocortical microcircuits in the mouse primary visual cortex (layers 2 to 3) to examine how four major interneuron classes-Parvalbumin, Somatostatin, vasoactive intestinal peptide, and LAMP5-interact within a competitive-cooperative motif to implement soft winner-take-all (sWTA) circuit dynamics. In a conductance-based network grounded in in vitro physiology, we show that this circuit motif selectively amplifies strong inputs while suppressing, without silencing, weaker ones, enabling gain modulation, signal restoration, and context-dependent multistability. Using a gain-matching strategy, we mapped the motif onto IBM's TrueNorth neuromorphic chip, revealing a principled correspondence between cell-type-specific roles and hardware primitives. Sparse coupling of sWTA modules generated persistent up-states and a two-state neural machine approximating working memory. We then embedded the sWTA circuit as a preprocessing filter in a Vision Transformer, which significantly enhanced out-of-distribution generalization across diverse tasks, including zero-shot digit classification, cross-domain transfer between digit datasets, and nighttime semantic segmentation. The sWTA filter boosted accuracy on unseen data by up to ~20% and reduced training compute by directing learning toward salient features, without additional data or architectural changes. By unifying biophysically grounded circuit models, neuromorphic implementation, and state-of-the-art AI architectures, this work outlines a generalizable roadmap for embedding cortical computation into next-generation NeuroAI systems that combine biological principles with practical AI performance gains.