And the radio itself cuts out too? Not just its backlights but the radio too?

Normally I would say that the problem would have to be in the fusebox itself as the two circuits have nothing to do with each other but automatic headlights make this a bit more complicated. Since the control unit for the automatic headlights goes through the ignition switch then this makes a bad ignition switch (or a bad connection there) a possibility.

Try this... see if the headlights flicker off when they are manually turned on. If so then the problem has to be in the interior fuse box... or the wiring going to it. If not then the problem could very well be a bad ignition switch. Also a bad ground somewhere could cause similar issues but the headlight switch should bypass the lighting module.

Unfortunately electrical gremlins such as this can become very complicated and time consuming to diagnose, so don't be surprised if it ends up having to go to a professional.