Can you give a practical numerical example illustrating how bayesian p-values should be used for model checking?
I've seen you post some questions pretty frequently looking for guidelines (e.g. how do I know when this is good enough). That's great and an excellent attitude, but I think you're going to be frustrated with a lot of Bayesian Stats because the answer will more often than not be "it depends".
And this answer is no different. The posterior probability that you observe data at least as extreme as some value is not a hard and fast rule as it is in Frequentism. In my own experience, I don't use this quantity in isolation. I instead loop in subject matter experts and ask them targeted questions. For example, I might ask "how many patients out of 100 might have a max concentration: larger than $C$? The subject matter expert will give me some approximation which I can then use to check my models. But the important part here is that I could not know that number a priori, and so there is no "small enough" for every situation.
It heavily depends on what you're doing and what is important to you.