Where am I supposed to get you schematics from? It's a computer's power supply so there are none.
Input voltage shouldn't matter. It shouldn't matter what the input voltage is as long as the actual voltage of the cap remains at or below 2.7V correct?
Oh good grief. The schematic of your circuit, not the PSU internals.
If you haven't bothered to make any measurements of voltage or current, and have only tried a very limited range of resistor values, then you really are stabbing around in the dark.
For example, what makes you think the voltage across the capacitor did remain below 2.7V?
What was the peak current flow?
What was the ripple current specification of the capacitor.
What was the capacitor's ESR?
What happens if you try a 1kohm resistor?
What wattage do you expect the resistor to have to dissipate?