[Old Tele man]

...instead of a separate battery, you MIGHT be able to get away with using one of the readily available "super capacitors" commonly used by the BOOMBOX boys/girls.

...with the proper "sizing" (in Farads) the voltage will come-ON and fade-OUT like the interior lights do on newer cars (assuming the radio isn't one of those that needs a "keep-alive" voltage so it can remember station settings, etc.).