The Hella's (and the related HID components) were never the actual problem.
TL;DR - My Hella's are working again, and the problem wasn't the Hella's or the HID upgrade at all. It was my Bussmann relay/fuse block.
After my previous fiddling with the Hella's, I decided to take a closer look at the rest of the circuitry for them when I was creating wiring harness for the Diode Dynamics SS3 fog lights that I installed recently. As part of that, I removed the Bussmann relay/fuse box from the engine bay and carefully looked it over for loose connections, etc. I didn't find any, but it did seem like - perhaps - the connections between the relays and the Metri-Pack terminals that they seat into might have been less-than-ideal. As such, I pulled the connectors, inspected each one, and reinstalled them, adding some dielectric grease to each connector prior to reseating the relays (and fuses).
Since then, the Hella's have worked flawlessly - turning on and off when they should. My best guess is that over time, a bit of surface oxidation had taken place to the contacts in the Bussmann, and so current wasn't flowing like it should have.
I'm still not totally happy with the HID Hella's since they emit quite a bit of radio frequency interference (RFI), and putting some high-powered Diode Dynamics lights on the bumper is very much top of mind, but the issue of the Hella's failing to turn on is now resolved.
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