Let´s go at it then:

A child at school approaches the teacher saying: "Can I go to the bathroom?". To which the teacher replies: "I don't know, can you?", implying that he doesn't know whether the student is capable of going to the bathroom by himself or not. So the teacher goes on: "Maybe you meant 'May I go to the bathroom?'".

No, I am afraid you didn´t understand the example.

It´s not implied that the teacher is thinking about whether the student is capable of going to the bathroom by himself or not. It´s not about that at all.

In fact, the teacher is being sarcastic, ironic, when he rephrases the student´s question.

He suggests that the student addressed him not in proper ways (teacher being older and in higher social status), so he should be using "may I go to the bathroom"?

So, the teacher resorts to the other meaning of can (physical ability) by asking the question  "I don't know, can you?" 

You are right when you say that "can" to ask for permission, is common and acceptable among native speakers.

I hope it helps.