Huh? You can derive all of those from Bayesian models. If you're counting balls from a bin with replacement, and your bot has counted 400Red with 637Blue, you have a Beta/Binomial model. That means you p_blue | data ~ Beta(401, 638) assuming a Uniform prior. The probability of observing a red ball given the above p_blue | data is P(red_obs | p_blue) = 1 - P(blue_obs | p_blue), which is calculable from p_blue | data. In fact in this simple example you can even analytically derive all of these values, so you don't even need a simulation!

Suppose the likelihood it missclassified a ball is significantly different from zero, but not yet known precisely.

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.

Can't you just add that to your equation? Seems like for anything real, this will not go many levels deep at all before it's irrelevent.