Full-state quantum circuit simulation requires exponentially increased memory size to store the state vector as the number of qubits scales, presenting significant limitations in classical computing systems. Our paper introduces BMQSim, a novel state vector quantum simulation framework that employs lossy compression to address the memory constraints on graphics processing unit (GPU) machines. BMQSim effectively tackles four major challenges for state-vector simulation with compression: frequent compression/decompression, high memory movement overhead, lack of dedicated error control, and unpredictable memory space requirements. Our work proposes an innovative strategy of circuit partitioning to significantly reduce the frequency of compression occurrences. We introduce a pipeline that seamlessly integrates compression with data movement while concealing its overhead. Additionally, BMQSim incorporates the first GPU-based lossy compression technique with point-wise error control. Furthermore, BMQSim features a two-level memory management system, ensuring efficient and stable execution. Our evaluations demonstrate that BMQSim can simulate the same circuit with over 10 times less memory usage on average, achieving fidelity over 0.99 and maintaining comparable simulation time to other state-of-the-art simulators.