I figured out how to install it using Windows PE. First, get a Windows installer image or WADK and mount it. Microsoft provides these, but hides the download link well. Then get the ME firmware update installer and the AMT/ME software installer from Lenovo and execute the unpackers with wine. From ~/.wine/drive_c/DRIVERS/WIN/AMT, do "cabextract SetupME.exe". You can discard everything but the "HECI_REL" directory from this, including SetupME.exe.
Now you need wimlib to create the WinPE image: "mkwinpeimg --windows-dir=/mnt winpe.img --overlay=$HOME/.wine/drive_c/DRIVERS"
The resulting winpe.img can be dd'ed to a USB thumb drive. Boot into it, and execute "cd /WIN/AMT/HECI_REL/win10", "drvload HECI.inf" (to load the MEI driver) and then "cd /WIN/ME/", "MEUpdate.cmd" to update the ME firmware.
Now you need wimlib to create the WinPE image: "mkwinpeimg --windows-dir=/mnt winpe.img --overlay=$HOME/.wine/drive_c/DRIVERS"
The resulting winpe.img can be dd'ed to a USB thumb drive. Boot into it, and execute "cd /WIN/AMT/HECI_REL/win10", "drvload HECI.inf" (to load the MEI driver) and then "cd /WIN/ME/", "MEUpdate.cmd" to update the ME firmware.