I managed to launch it without installing a full DE, here is what I did in case anyone is interested (although installing a DE is probably the simplest option):

Install xorg and xinit:

sudo apt install -y xorg xinit

In /etc/X11/Xwrapper.config set allowed_users=anybody

Add the current user (or replace with user that launches the command) to these groups:

sudo usermod -aG input,tty,video $USER

Grant access to the second tty to current user (this need to be repeated after a reset):

sudo chown $USER /dev/tty2

Now replace the command to launch the emulator in /opt/retropie/configs/ps2/emulators.cfg:

pcsx2-nogui = "xinit /opt/pcsx2-unstable/pcsx2-qt -fullscreen -nogui %ROM% -- :0"

It's important to use the full path /opt/pcsx2-unstable/pcsx2-qt and not just pcsx2-qt because the later is actually a shell script that calls the former and xinit doesn't like that:

cat $(which pcsx2-qt) #!/usr/bin/bash /opt/pcsx2-unstable/pcsx2-qt "$@"

And that's it. Launching a game will show the PCSX2 gui where we can choose the different folders for roms and BIOS. This step require a keyboard and ideally a mouse.

Runcommand controller Issues