Future wireless networks are envisioned to employ multiple-input multiple-output (MIMO) transmissions with large array sizes, and therefore, the adoption of complexity-scalable transceiver becomes important. In this paper, we propose a novel complexity-scalable transceiver design for MIMO systems exploiting bit-interleaved coded modulation (termed MIMO-BICM systems). The proposed scheme leverages the channel bidiagonalization decomposition (CBD), based on which an optimization framework for the precoder and post-processor is developed for maximizing the mutual information (MI) with finite-alphabet inputs. Particularly, we unveil that the desired precoder and post-processor behave distinctively with respect to the operating signal-to-noise ratio (SNR), where the equivalent channel condition number (ECCN) serves as an effective indicator for the overall achievable rate performance. Specifically, at low SNRs, diagonal transmission with a large ECCN is advantageous, while at high SNRs, uniform subchannel gains with a small ECCN are preferred. This allows us to further propose a low-complexity generalized parallel CBD design (GP-CBD) based on Givens rotation according to a well-approximated closed-form performance metric on the achievable rates that takes into account the insights from the ECCN. Numerical results validate the superior performance of the proposed scheme in terms of achievable rate and bit error rate (BER), compared to state-of-the-art designs across various modulation and coding schemes (MCSs).