In recent years, the introduction of neural networks (NNs) into the field of speech enhancement has brought significant improvements. However, many of the proposed methods are quite demanding in terms of computational complexity and memory footprint. For the application in dedicated communication devices, such as speakerphones, hands-free car systems, or smartphones, efficiency plays a major role along with performance. In this context, we present an efficient, high-performance hybrid joint acoustic echo control and noise suppression system, whereby our main contribution is the postfilter NN, performing both noise and residual echo suppression. The preservation of nearend speech is improved by a Bark-scale auditory filterbank for the NN postfilter. The proposed hybrid method is benchmarked with state-of-the-art methods and its effectiveness is demonstrated on the ICASSP 2023 AEC Challenge blind test set. We demonstrate that it offers high-quality nearend speech preservation during both double-talk and nearend speech conditions. At the same time, it is capable of efficient removal of echo leaks, achieving a comparable performance to already small state-of-the-art models such as the end-to-end DeepVQE-S, while requiring only around 10 % of its computational complexity. This makes it easily realtime implementable on a speakerphone device.