Thank you TK and gchart. Agreed that there doesn’t look like there’s any place on the component side for a bleeder resistor.
So I tried different resistors on the spring side. First I tried a 680 ohm resistor and got the same result as the 328 ohm resistor. Then I tried a 220 ohm resistor and BAM, SUCCESS! Here’s my set-up. I didn’t want to put everything together to test each resistor, so I wired it up manually.

It was a challenge to stretch the solder from the spring to the resistor and the resistor to the retaining ring, but eventually I got it to work. To clean things up, I put the driver onto a drill and sanded down the solder that went onto the retainer ring.

I kept the 2K resistor on the switch and replaced the original LED with 2 0805 blue LEDs. I measured 0.56 mA for the tailcap.

It looked fine before, but now that it has 2 LEDs it looks so much better. I feel proud that I was able to make this work.