Deploying object detection models for resource-constrained remote sensing applications necessitates on-board model inference capabilities. While Field Programmable Gate Arrays (FPGAs) offer massive parallelism as energy-efficient hardware platforms, model quantization remains essential to further balance computational efficiency with detection accuracy. Compared to post-training quantization methods that involve multiple-stage development with consistent dependency on domain datasets, quantization-aware training (QAT) integrates quantization constraints into training, providing a simpler pipeline for model compression. However, QAT introduces quantization errors to which smaller objects are more vulnerable. To address this issue, we propose object-scale-aware (OSA) regularization that amplifies quantization error penalties for smaller targets. Our approach is validated through two case studies: bird detection at airports and aerial-view building detection. We perform 8-bit QAT on YOLOX series models using the MVA2023 dataset and the Bavarian Building Dataset for the respective studies. Our method achieves up to 50.2 times inference acceleration with minimal accuracy loss on Xilinx Kria KV260 FPGAs compared to full-precision models. The ablation study and detection examples further demonstrate the effectiveness of OSA regularization in small object detection.