THE BATTERY IS GOOD BUT THE RADIO CUTS OUT AFTER ABOUT 5 MINUTES OF BEING STATIONARY - I HAVE TO TURN ON THE ENGINE TO BRING THE RADIO ON AGAIN?
In ready or in ACC? If it's in ready and stationary. So radio operation some how linked to engine? Maybe it could be a bad connection somewhere? And it changes when car shakes with engine idling. You have no fault lights or codes? If it's only in ACC. How did you check that 12v battery is good? Depending on method maybe you have a bad connection or a bad battery. Meansure the voltage at jump start terminal when the radio is working. Is the voltage dropping before radio shuts down?
If you're listening to it in ACC or ON ,,, STOP THAT! Seriously, that's a really bad idea in a Prius. Keep the car in READY mode if you're going to listen to the radio. That dinky 12V aux battery is not like the one in your neighbor's diesel F350. It's just made to fire up some computers and close a couple relays. As a result, newer models of Prius will shut off after a few minutes in ACC or ON to protect the aux battery.
I see it's a UK shreadout. Will a 2006 model across the pond have this? My 2008 USA brand would run that 12V flat if given half a chance.