With six LEDs, I'll bet three of the 2.5F caps will get it done for two minutes. 
Even if they get a bit dimmer, they're still light, and the slow decay would probably not be that noticeable. It's simple and fairly cheap to build.
I stand ready to throw darts at any circuit you propose! 
In your previous example of the supercap dropping to 4V, I calculated an approximate drop of 30-40% in LED brightness. This is occurring over many tens of seconds you so probably don't notice it. And it does make for a simple circuit. But if the OP wants constant-brightness, then additional regulation is required.
However, I'm curious how you make a simple and cheap rapid charging circuit. A 1 Amp constant-current charger that cuts off when the supercap(s) reaches 5V is "easy" if you have a bench power-supply. But a hand-built circuit?
Also, when the supercap is charging for tens of seconds, I'd think you want full-brightness on the LEDs. Or maybe not? If so you need a bypass circuit - could be as simple as diode OR'ing - to power the LEDs to full-brightness whenever track power is available.
