How to ask for approval letter from dean?
I don't know anything about your university or its normal practices, so this is only tentative. 

If your university has any sort of office for student advocacy, go to them with your concerns. Hidden rules aren't appropriate in any situation, though they may be permitted - unfortunately. 

In case there is no such office, go to any professor that you trust and who knows you well, and ask him/her to be your advocate. The dean will certainly offer a bit of time to a professor that they won't to you. You are asking for an exception to the "rules". Make sure that you have a basis for the exception (you have the needed knowledge to proceed) and not just that the rule is unfair if it is hidden. 

Unfortunately, you are possibly making people uncomfortable and when that happens they sometimes behave badly. The more confrontational you make it, the worse it may get, so be aware of the risks.