Thanks, but that wouldn't work because a ME with a borked firmware package won't accept a driver. A driverless ME can't have the firmware updated.
What ended up happening was my board had some kind of bug in the UEFI BIOS that caused it to freeze during firmware backup (this board has two chips to hold BIOS ROMs - a primary and a secondary to act as a failsafe backup). I had to restart it when it was messing with its own BIOS package, which is a big no-no.
I guess what happens during that process is that the ME gets updated via a microcode flash at the END of the BIOS update process AFTER the system has been restarted. When that didn't happen because I had to hard reset it, the ME kind of got stuck in this weird state where it was forever waiting for that secondary update process to begin. This is also why simply installing different versions of my BIOS wasn't change the ME version; the ME was already stuck in that "waiting for update" phase, so the board telling it to enter that phase again wasn't returning the expected acknowledgement and the BIOS installer basically said "Okay, whatever" and skipped the process altogether.
I basically figured out that pulling your CMOS battery, pulling the plug on your PSU, and waiting 10 minutes or so for the board capacitors to drain their residual voltage is enough to clear from the MEs volatile memory whatever data entry kept causing it to enter that update phase each power cycle. I could then install another BIOS version, the ME updated as planned, and everything was fine.
To anyone having this problem, that's what ya gotta do. Good luck!