Conventional RGB-based high dynamic range (HDR) imaging faces a fundamental trade-off between motion artifacts in multi-exposure captures and irreversible information loss in single-shot techniques. Modulo sensors offer a promising alternative by encoding theoretically unbounded dynamic range into wrapped measurements. However, existing modulo solutions remain bottlenecked by iterative unwrapping overhead and hardware constraints limiting them to low-speed, grayscale capture. In this work, we present a complete modulo-based HDR imaging system that enables high-speed, full-color HDR acquisition by synergistically advancing both the sensing formulation and the unwrapping algorithm. At the core of our approach is an exposure-decoupled formulation of modulo imaging that allows multiple measurements to be interleaved in time, preserving a clean, observation-wise measurement model. Building upon this, we introduce an iteration-free unwrapping algorithm that integrates diffusion-based generative priors with the physical least absolute remainder property of modulo images, supporting highly efficient, physics-consistent HDR reconstruction. Finally, to validate the practical viability of our system, we demonstrate a proof-of-concept hardware implementation based on modulo-encoded spike streams. This setup preserves the native high temporal resolution of spike cameras, achieving 1000 FPS full-color imaging while reducing output data bandwidth from approximately 20 Gbps to 6 Gbps. Extensive evaluations indicate that our coordinated approach successfully overcomes key systemic bottlenecks, demonstrating the feasibility of deploying modulo imaging in dynamic scenarios.