This thread of Nick Gammon has me thinking about capacitors, specifically supercaps although the one in his example is a normal one I think.
A 1F supercap like this is less than 20mm diameter.
Could that play a part in your solution?
Boy, things have changed, the line printers we worked on we're 1F in a 36"x36"x36" package, they were 36 volts however. Some melted screw drivers on it.