So, I am designing a door authentication system (can't really go into more detail) for our school, so that only authenticated persons can go through a certain internal door. They hold that its inner working should be kept a secret, so that no one can reverse engineer it. I maintain that this would be Security through obscurity, which is bad. I have designed the system such that knowing how it works wouldn't help you get in, only having the key would help you get in, as according to Kerckhoff's principle. They still maintain that instead of more people knowing about it, less should.

Is it a good idea to go with a closed design? If not, exactly why?

P.S. They only told me it was a secret after I had designed most of it and told a bunch of people, if that makes a difference (I had no idea they wanted it to be a secret).
P.P.S. Although it is not open to the outside, the room it is protecting contains more valuable stuff than the rest of the school, so I would expect adversaries to be able to reverse engineer it anyways if we go with a closed design instead of an open one, but I'm not sure how to communicate that.

Obscurity isn't a bad security measure to have in place. Relying upon obscurity, as your sole or most substantial security measure, is more or less a cardinal sin.

Kerckhoff's Principle is perhaps the most oft-cited argument against implementing security through obscurity. However, if the system is already properly secure according to that principle, this is mis-guided.

The principle, paraphrased, says "assume the enemy knows everything about the design of your security systems". This does not in any way imply "tell the enemy everything about your system, because they know anyway".

If the system is already well-designed according to Kerckhoff's Principle, the worst that can be said about applying security through obscurity to it is that it adds little to no value. At the same time, for such a system, there is equally little to no harm in protecting the confidentiality of its design.
