Conventional audio coding technologies commonly leverage human perception of sound, or psychoacoustics, to reduce the bitrate while preserving the perceptual quality of the decoded audio signals. For neural audio codecs, however, the objective nature of the loss function usually leads to suboptimal sound quality as well as high run-time complexity due to the large model size. In this work, we present a psychoacoustic calibration scheme to re-define the loss functions of neural audio coding systems so that it can decode signals more perceptually similar to the reference, yet with a much lower model complexity. The proposed loss function incorporates the global masking threshold, allowing the reconstruction error that corresponds to inaudible artifacts. Experimental results show that the proposed model outperforms the baseline neural codec twice as large and consuming 23.4% more bits per second. With the proposed method, a lightweight neural codec, with only 0.9 million parameters, performs near-transparent audio coding comparable with the commercial MPEG-1 Audio Layer III codec at 112 kbps.