So I tried to use win-to-go usb stick to run Win 10 out of it and execute that hdmi patch from Intel (as this exec is running only from Win 10) - hoping that this approach will not jeopardize my LE setup there.
What a pain! Tried with 32 and 64 bits Win 10 recovery disk, then official Win 10 iso images (again 32 and 64 bits) which Rufus can use to build win-to-go usb stick - and neither worked properly, they did not want to boot or were crashing soon after booting. Finally it booted nicely from HirenPE, and then (obviously) Intel tools started failing, I suppose as Hiren is missing specific NUC drivers.
So here is my collection so far - screenshot and the log file:
Gfx Selection = 0
Graphics adapter initialization - NVIDIA: failed! (-2)
Graphics adapter initialization - AMD: Loading DLL failed!
Graphics adapter initialization - Intel (LS-PCON): failed! (-2147221164)
Graphics adapter initialization failed!
[Chip Information]
Reset Bobcat failed!
Hiren seems to have option to add custom drivers, but the problem with NUC is that most of these drivers are in a form of exe file which just need to be executed on the device. Ughh.