The 1A supply is actually easier than it sounds. You just need the 5.5 V adjustable supply, the diode, and a low value resistor. I'd consider a 5-6 ohm to start. Dropping 5V across the 6 ohm during the initial charging state is possible and it ends up less than an amp surge current. It actually charges the caps pretty quickly, and remember that we're not ever totally discharging them, we just want to run 2 minutes and then we'll have the "charger" working again. When we're running under power, the few ohms of the series resistor is insignificant in lighting the LED's.
The only time you'd ever notice the caps charging is on initial power on, after that they should already be charged to a reasonable level.
It's the KISS method. No switching, just a 1A adjustable supply, a diode, and a couple of selected resistors.
Fire those darts, be make sure you hit the target. 
Now, if he's anal about truly constant intensity, you'd probably have to add some complexity. Truthfully, I think this would do the trick and nobody that didn't know the secret would realize that the intensity changes some during the power off cycle.
KISS is fine, but a fixed-value resistor is a very slooow way to charge a supercap. So when you say 5 ohm resistor, that is a peak current of 1 Amp if from a 5V source and only occurs if the supercap starts totally discharged. When the supercap reaches 3V, the charging current drops to 0.4 Amps. When the supercap reaches 4V, the charging current drops to 0.2 amps. And the current keeps dropping. Basically you have the so-called RC time-constant which is the product of R and C, or in this case 5 ohms times 2.5 Farads which is 12.5 seconds. Starting from a discharged cap, it takes about 4 time constants or 50 seconds to charge up the cap to 99%. OTOH, if you had a constant-current charger which is what you'd find in engineered rapid-charger circuits (more complex), a 1 Amp charger would fully charge the cap 4 times faster.
So with your basic KISS circuit, the LEDs would slooowly dim over a minute or whatever when no power is available, and then slooowly get brighter over a minute or whatever when power is restored. It's not a question of right or wrong but just need to be clear on the alternatives.
