This research presents a novel framework integrating Flexible-Duplex (FlexD) and Integrated Sensing and Communications (ISAC) technologies to address the challenges of spectrum efficiency and resource optimization in next-generation wireless networks. We develop a unified system model for a dual-functional radar-communication base station with multiple-input multiple-output capabilities, enabling dynamic uplink and downlink channel allocation. The framework maximizes network throughput while maintaining radar sensing performance, subject to signal-to-clutter-plus-noise ratio (SCNR) requirements and power constraints. Given the non-convex and combinatorial nature of the resulting optimization problem, we propose an iterative algorithm that converges to a locally optimal solution. Extensive simulations demonstrate the superiority of the proposed FlexD-ISAC framework compared to conventional half-duplex networks. Additionally, sensitivity analyses reveal the impact of SCNR requirements and power constraints on system performance, providing valuable insights for practical implementation. This work establishes a foundation for future research in dynamic, resource-efficient wireless systems that simultaneously support sensing and communication capabilities.