It sounds like your ideas are potentially dangerous, and probably illegal?.
A more sensible solution would be to have an audio indication that sounds after x seconds while the indicators are on, and repeats every y seconds if they continue to be on. This would require no real modifications to the car, and could be accomplished inside the passenger compartment, requiring no special precautions and only low power consumption.
However, by far the easiest solution would be a microcontroller, and count the number of flashes.
I'm a bit puzzled what you're doing to make a 556 overheat?, please post your schematic.