Keeping a 12 V circuit live briefly
We know dV/dt = Ic/C  and you said dt= few seconds but we don’t know allowed droop or dV.  Let’s assume that the tolerance is 10% or 0.1 * 12V or 1.2V droop. 

Then for dt=3s and 6W =12V*0.5A.  = P = V * I or I=P/V

\$C=P/V * dt/ dV= \dfrac{P}{V^2*0.1}\$

So C=0.4 Farad which is pretty big but based on my assumptions

Now for a 12V source to a 470R and 3V LED @20mA you can get 10mA at 7.7V and Pavg is 10V*15mA =150’mW And Pmax is 12*20mA so for dt=1s , dV =1/3 of 12V or 4V, C =3,000 uF