Assume:
-. Test current = 500mA
-. LED headlight takes below 500mA to lit, and drops 10V.
Approximated voltage and time correlation is: dV = 1/C * I * dt
To hold voltage below 10V for 10 seconds: C = (500mA * 10sec) / 10V = 0.5F
So, combination of super capacitors with 0.5F/12V may do the job, for this hypothetical case. Meantime, I thought, LED headlights use inline converter(voltage boost/current regulation). If so, the story will be different.
Edit
I would feel sorry, if I don't tell this now; You will need a few tons of fuses, once you start testing with the capacitor.