Go for the "bootable CD" option, if it's available. You don't need Windows for that. My ThinkPads all run Linux and I have no problems updating them.
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.