Running offers substantial health benefits, but improper gait patterns can lead to injuries, particularly without expert feedback. While prior gait analysis systems based on cameras, insoles, or body-mounted sensors have demonstrated effectiveness, they are often bulky and limited to offline, post-run analysis. Wrist-worn wearables offer a more practical and non-intrusive alternative, yet enabling real-time gait recognition on such devices remains challenging due to noisy Inertial Measurement Unit (IMU) signals, limited computing resources, and dependence on cloud connectivity. This paper introduces StrikeWatch, a compact wrist-worn system that performs entirely on-device, real-time gait recognition using IMU signals. As a case study, we target the detection of heel versus forefoot strikes to enable runners to self-correct harmful gait patterns through visual and auditory feedback during running. We propose four compact DL architectures (1D-CNN, 1D-SepCNN, LSTM, and Transformer) and optimize them for energy-efficient inference on two representative embedded Field-Programmable Gate Arrays (FPGAs): the AMD Spartan-7 XC7S15 and the Lattice iCE40UP5K. Using our custom-built hardware prototype, we collect a labeled dataset from outdoor running sessions and evaluate all models via a fully automated deployment pipeline. Our results reveal clear trade-offs between model complexity and hardware efficiency. Evaluated across 12 participants, 6-bit quantized 1D-SepCNN achieves the highest average F1 score of 0.847 while consuming just 0.350 microjoule per inference with a latency of 0.140 ms on the iCE40UP5K running at 20 MHz. This configuration supports up to 13.6 days of continuous inference on a 320 mAh battery. All datasets and code are available in the GitHub repository https://github.com/tianheng-ling/StrikeWatch.