Overfitted neural image codecs offer a promising path to ultra-low-complexity decoding, ideal for resource-constrained hardware. However, realizing their potential requires an integer-only pipeline, a task complicated by their extreme sensitivity to post-training quantization (PTQ). To our knowledge, this is the first work to systematically address the challenges of 8-bit PTQ for such codecs. We present an effective methodology for 8-bit weight, 16-bit activation quantization, developed through a three-stage investigation. (1) We first establish that a fine-grained, per-channel quantization scheme is the essential foundation for preserving model fidelity. (2) We then use Hessian analysis to identify strong weight dependencies as the primary bottleneck and apply a dependency-aware compensation algorithm to mitigate the resulting errors. (3) To further refine the compensation process, we integrate a principled, decreasing-importance order into the quantization sequence. Our final approach reduces the rate-distortion overhead to just +2.95% BD-Rate on the Kodak dataset and +2.78% BD-Rate on the CLIC dataset, establishing a practical methodology for this sensitive codec class.