Will,
As I have doubts that changing the pull-up resistor on RESET will suffice, I gave the problem a bit more thought. How about paralleling the supply of that resistor (or the pin itself) from the regulator output with a supply from the incoming 12 V (using a suitable voltage divider) from the PC’s power supply?
If you put an appropriate diode between the voltage divider and the RESET pin, that should preclude backfeeding from the capacitors to the PC’s supply. Putting a diode in the connection between the capacitors and the RESET pin will keep the 12 V power line from trying to charge the capacitors through the voltage divider. To ensure you’re getting 5 V to the pin, I suggest you select the resistors for the divider so that the voltage after the diode’s forward drop is 5 V. (If you’d like to take advantage of a “freebie”, you can use an LED to prevent backfeeding while providing indication that the PC power supply is on. To do that, though, you’ll need to put the LED between incoming power and the voltage divider, so that there’s a path to ground.)
Eric