In wireless acoustic sensor networks (WASNs) with high spatial resolution, energy efficiency is crucial for speech enhancement (SE) due to limited node energy. To optimize energy efficiency under predefined performance constraints, sensor selection (SS), rate allocation (RA), or a joint RA and SS (RASS) approach can be employed. However, since the above criteria were developed in a frequency-dependent manner, it may lead to conflicts among SS results over different frequencies. In this work, we propose a novel RASS model by promoting bit rate sparsity in RA, eliminating NP-hard Boolean programming, and extending it to a frequency-invariant RASS (FI-RASS) for consistent SS results across frequencies. In addition, we introduce a re-weighting strategy to further reduce energy consumption in the WASN. Compared with existing RA, SS, and RASS approaches, the FI-RASS provides a remarkable superiority in terms of energy efficiency. Numerical experiments demonstrate the effectiveness of the proposed method.