Most times (if we're talking business-class laptops/desktops here), you can update the firmware independently from the BIOS/UEFI firmware. The ME firmware sits on a separate region of the flash.
But Intel doesn't seem to be releasing the firmware on its websites ... they only have a flasher, and the firmware comes from OEM. Damn, wly the hell?
At least, if you're comfortable soldering chips, you can try me_cleaner or somthing.
Edit: https://www.win-raid.com/t596f39-Intel-Management-Engine-Drivers-Firmware-amp-System-Tools.html
This site seems to provide separate F/W packages for all versions of MEI.
However, what if there are remotely exploitable zero-days? (I think that there ARE ones in the wild, which may be why the US Gov't. asked for HAP support on Skylake MEI and above)