I have the same setup (EndeavourOS / KDE plasma 6 / Wayland / SDDM / 2 monitors) and had the same problem. The worst thing is that typing the password in the “active” login prompt (the one with the focus) wasn’t working anyway, so I had to use the mouse to give focus to the other monitor first, and then type the password. Absolutely annoying.

The solution I found (sorry I forgot where, some forum) is to disable all the detected monitors except one in /usr/share/sddm/scripts/Xsetup. Basically your secondary monitor will not get any signal until you type the password and log in. At that point any other monitor will be reactivated automatically.

This is my Xsetup:

#!/bin/sh
xrandr --output DisplayPort-0 --mode 2560x1440 --pos 0x0 --rotate normal
xrandr --output DisplayPort-1 --off
xrandr --output HDMI-A-0 --off
xrandr --output HDMI-A-1 --off

IMPORTANT

Check out the output of xrandr in Wayland on my system:

$ xrandr | grep ' connected'
DP-1 connected primary 2560x1440+0+0 (normal left inverted right x axis y axis) 590mm x 334mm
DP-2 connected 2560x1440+2560+0 (normal left inverted right x axis y axis) 590mm x 334mm

DP-1 and DP-2 are the names used by Wayland, but they don’t work in Xsetup because X11 calls the ports DisplayPort-0 and DisplayPort-1 - and I don’t remember if HDMI ports are also called differently.

So you need to log in X11 first, get the names with xrandr, create or update Xsetup and reboot.

Which mini pc? I have an Intel NUC Intel i5 and looking for something smaller but can run a dev desktop (xfce, vscode, node, docker).