Supercapacitor Bank Wont Charge w RPi Load
The 1 ohm balancing resistors are completely inappropriate. Without the capacitor spec sheets, I would estimate their values should be north of 10k each. They may not even be necessary given the 2.7 volt rating of the caps. As they are, the capacitors will never correctly charge and the 3 ohm series string is a substantial load to the supply. Hopefully this is a drafting error that you can comment on and correct.

The negative lead of the cap bank is isolated from the 5.35 volt negative. This is not correct. It should be tied to the negative rail.

I would also be concerned about the surge current when the caps are largely depleted. At this point, they are effectively a short circuit to the USB supply. This may be what you are seeing. Try placing a 10 ohm or so resistor in series with the cap bank to see if this corrects your problem. This will increase the charge time but limit the initial charge surge.