In this work, we expand the cooperative multi-task semantic communication framework (CMT-SemCom) introduced in [1], which divides the semantic encoder on the transmitter side into a common unit (CU) and multiple specific units (SUs), to a more applicable design. Our proposed system model addresses real-world constraints by introducing a general design that operates over rate-limited wireless channels. Further, we aim to tackle the rate-limit constraint, represented through the Kullback-Leibler (KL) divergence, by employing the density ratio trick alongside the implicit optimal prior method (IoPm). By applying the IoPm to our multi-task processing framework, we propose a hybrid learning approach that combines deep neural networks with kernelized-parametric machine learning methods, enabling a robust solution for the CMT-SemCom. Our framework is grounded in information-theoretic principles and employs variational approximations to bridge theoretical foundations with practical implementations. Simulation results demonstrate the proposed system's effectiveness in rate-constrained multi-task SemCom scenarios, highlighting its potential for enabling intelligence in next-generation wireless networks.