If you use a model that doesn't ask you to think about this likelihood at all, you will get the same result as if you had used bayes and consciously chose to approximate the likelihood of misclassification as zero.

You may get slightly better results if you have a reasonnable estimate of that probability, but you will get no worse if you just tell Bayes zero.

It feels like you're criticizing the model for asking hard questions.

I feel like explicitely not knowing an answer is always a small step ahead of not considering the question.