I'm not sure the conflict is exactly between epistemic and instrumental rationality. That would be when believing falsely is instrumentally rational. In this case, one action (concealing ignorance) is both good for one instrumental subgoal (status/positive evaluation) and bad for another (epistemic rationality). The epistemic problem does not lead to benefit; doing something beneficial leads to it.
Here's it phrased another way: In a case of actual conflict, the benefit comes from the absence of true knowledge or the presence of false knowledge, so if one magically learned the truth with nothing else changing, this would negate the instrumental benefit. In the case of concealing ignorance, if one magically learned the information without the tutor knowing of the earlier ignorance this would be even better instrumentally. The ignorance is not necessary for the instrumental benefit, only the tutor's awareness of it.
Reading Being a teacher made me think about my experiences tutoring university students. I'm a PhD student, so my teaching currently consists of helping first year undergraduates on problem sheets. I think I'm reasonably good- I try to appreciate what difficulties they are having and anticipate them, by explaining what they are doing means, and approaching the problem in different ways.
One constant frustration I get though is that, having explained a problem to a particular student, the student will give me a blank look, and then mutter "ok". I know what that look means, and will ask "so do you understand that?" "sort of...." "well look at it this way....".
Now some of this may come from me- I'm explaining too fast or in a way they don't understand, and my familiarity with the subject, but I suspect some comes from them. It can be difficult to admit one's ignorance, from my own experience. I, and I suspect others who go on to do university maths, was used to being the best or near the best in school, with "being smart" being part of my core identity, something that made me distinct from my more attractive or more fit peers. Getting to university and realising one is having difficulty with even basic questions can be a knock to ones identity. So I hid my ignorance, and did myself damage. I might lose the respect of a tutor, or even a lecturer by admitting my ignorance, but the alternative is to remain ignorant.
I suspect this is a problem that is common among us all. Its a lot easier to pretend we understand, and sometimes it may help- if we want to impress a potential employer we shouldn't admit ignorance (unless the alternative makes us look more ignorant)- but in general admitting ignorance helps us learn. There is (almost) always someone with more knowledge on a particular subject than you, and a failure to use that resource is a failure of rationality.