Inverse protein folding, the process of designing sequences that fold into a specific 3D structure, is crucial in bio-engineering and drug discovery. Traditional methods rely on experimentally resolved structures, but these cover only a small fraction of protein sequences. Forward folding models like AlphaFold offer a potential solution by accurately predicting structures from sequences. However, these models are too slow for integration into the optimization loop of inverse folding models during training. To address this, we propose using knowledge distillation on folding model confidence metrics, such as pTM or pLDDT scores, to create a faster and end-to-end differentiable distilled model. This model can then be used as a structure consistency regularizer in training the inverse folding model. Our technique is versatile and can be applied to other design tasks, such as sequence- based protein infilling. Experimental results show that our method outperforms non-regularized baselines, yielding up to 3% improvement in sequence recovery and up to 45% improvement in protein diversity while maintaining structural consistency in generated sequences.