I would try a resistor in series to the 7805, I have used really small and cheap ones from Mouser. Usually 1 watt, they are like 0.4" long and $0.10.
Up the resistance until the voltage at the 7805 meets requirements. Probably 8V will be enough. Or put a 7809 in front?
The other thing, put a TO220 big regulator on it. Also, if you have an LCD with back light, connect it to 12V thru a bigger resistor to take
the current out of the small regulator. That is usually 50 ma or so. You could maybe add a second regulator just for the LED back light, it is current you are setting, not voltage.
The switching version my not be good in a radio! May have some nice birdies!
I never use the regulator built into Nano Arduinos, it is so cheap to put a TO-220 one external?
I have blow up the ones on the Nanos, Nano is my favorite, I have used several of them in projects. eBay for a few bucks.