I've never messed with one of those before, so I don't know how much help I would be. I do know the noise your talking about, I have an old timex indiglo alarm clock that does it when you hit the night light button. From the reading I did on it, it seems that you are kind of stuck because of the need of the panel to bump up the power to light it, which is where the noise is coming from.
It does look nice and simple though, too bad it's loud.