Kodi only searches for the active DRM device at startup, so if you connect (or power on) the monitor afterwards the kernel will detect it and reconfigure itself for the new active connection but Kodi will remain fixed on whatever it saw at boot. And it looks like the DP-1 port is seen to be active when HDMI is not connected.
The getedid script should work, but you will need to ensure HDMI is connected and the monitor powered on before you power on the NUC, else the script will probably find the DP-1 device active and then you get the EDID for the DP-1 port, maybe?
It's also possible to set "video=HDMI-A-1:1920x1080M@60" in kernel boot params, which should force output to the HDMI connector with some standard timings. Other random thoughts are to disable the DP ports in BIOS, if that's possible?