A hybrid detector that fuses both quantized and full-precision observations is proposed for weak signal detection under additive and multiplicative Gaussian noise. We first derive a locally most powerful test (LMPT)--based hybrid detector from the composite probability distribution of the compound observations received by the fusion center, and then analyze its asymptotic detection performance. Subsequently, we optimize the sensor-wise quantization thresholds to achieve near-optimal asymptotic performance at the local sensor level. Moreover, we propose a mixed-integer linear programming approach to solve the optimization problem of transmission bandwidth allocation accounting for bandwidth constraints and error-prone channels. Finally, simulation results demonstrate the superiority of the proposed hybrid detector and the bandwidth allocation strategy, especially in challenging error-prone channel conditions.