Look to see if the initial torquing-over of the engine is moving a connector and breaking contact for a moment. How long does this 'radio-off' issue last? Also, look at the voltage with a voltmeter when this happesn and see if it spikes up or down.
I'll take a look around, for anything that looks like it might be getting strained. It only lasts about 1 second just enough for the radio to turn off. It comes back on that quick. Im taking the car for a drive tomorrow so I'll pay attention to the voltage to see what it does.