Your solution may very well also work, good thinking. A few things to consider though:

• Putting capacitors in series requires balancing them
• A silicon full bridge rectifier drops ~1.4V frm the input
• Incandecent bulbs require a lot of current to get going, this may result in a delay for them to turn on

I’ve actually done that with a simple single diode rectifier and a supercap with zener voltage limitation:

If your supercap has a too low series resistance for the LED to light immediately you need an additional resistor in series with it. It is simple and small enough to directly fit inside the original lamp casing and has been working without a problem for over 3 years.