On-device skin lesion analysis is constrained by the compute and energy cost of conventional CNN inference and by the need to update models as new patient data become available. Neuromorphic processors provide event-driven sparse computation and support on-chip incremental learning, yet deployment is often hindered by CNN-to-SNN conversion failures, including non-spike-compatible operators and accuracy degradation under class imbalance. We propose QANA, a quantization-aware CNN backbone embedded in an end-to-end pipeline engineered for conversion-stable neuromorphic execution. QANA replaces conversion-fragile components with spike-compatible transformations by bounding intermediate activations and aligning normalization with low-bit quantization, reducing conversion-induced distortion that disproportionately impacts rare classes. Efficiency is achieved through Ghost-based feature generation under tight FLOP budgets, while spatially-aware efficient channel attention and squeeze-and-excitation recalibrate channels without heavy global operators that are difficult to map to spiking cores. The resulting quantized projection head produces SNN-ready logits and enables incremental updates on edge hardware without full retraining or data offloading. On HAM10000, QANA achieves 91.6% Top-1 accuracy and 91.0% macro F1, improving the strongest converted SNN baseline by 3.5 percentage points in Top-1 accuracy (a 4.0% relative gain) and by 12.0 points in macro F1 (a 15.2% relative gain). On a clinical dataset, QANA achieves 90.8% Top-1 accuracy and 81.7% macro F1, improving the strongest converted SNN baseline by 3.2 points in Top-1 accuracy (a 3.7% relative gain) and by 3.6 points in macro F1 (a 4.6% relative gain). When deployed on BrainChip Akida, QANA runs in 1.5 ms per image with 1.7 mJ per image, corresponding to 94.6% lower latency and 99.0% lower energy than its GPU-based CNN implementation.