As a crucial part of the spectral filter array (SFA)-based multispectral imaging process, spectral demosaicing has exploded with the proliferation of deep learning techniques. However, (1) bothering by the difficulty of capturing corresponding labels for real data or simulating the practical spectral imaging process, end-to-end networks trained in a supervised manner using simulated data often perform poorly on real data. (2) cross-camera spectral discrepancies make it difficult to apply pre-trained models to new cameras. (3) existing demosaicing networks are prone to introducing visual artifacts on hard cases due to the interpolation of unknown values. To address these issues, we propose a hybrid supervised training method with the assistance of the self-supervised generative model, which performs well on real data across different spectral cameras. Specifically, our approach consists of three steps: (1) Pre-Training step: training the end-to-end neural network on a large amount of simulated data; (2) Pseudo-Pairing step: generating pseudo-labels of real target data using the self-supervised generative model; (3) Fine-Tuning step: fine-tuning the pre-trained model on the pseudo data pairs obtained in (2). To alleviate artifacts, we propose a frequency-domain hard patch selection method that identifies artifact-prone regions by analyzing spectral discrepancies using Fourier transform and filtering techniques, allowing targeted fine-tuning to enhance demosaicing performance. Finally, we propose UniSpecTest, a real-world multispectral mosaic image dataset for testing. Ablation experiments have demonstrated the effectiveness of each training step, and extensive experiments on both synthetic and real datasets show that our method achieves significant performance gains compared to state-of-the-art techniques.