My understanding is that lutris is a program whereas as Plasma is a desktop environment

Yes, that's right.

so I guess I would be looking for plasma to start regardless and if connected to a ps5 controller then open lutris to big picture mode.

Plasma (like many other desktop environments) can automatically launch whatever you like when you log in, if you create an XDG Autostart file for it in the $HOME/.config/autostart/ directory. (Hint: It uses the .desktop file format.)

You could write a script that checks for a Playstation controller and launches Lutris if one is detected. You could then reference that script in the autostart file's Exec= line.

The simplest way I know of to detect a Playstation controller in a shell script would be something like this:

#!/bin/sh
udevadm info -t |grep -q DRIVER=playstation && echo "Hello, Playstation!"

You would have to make the script executable, and replace the echo command at the end with a lutris command line, of course.

Terminology correction to your original post: Booting is something that a computer does when you power it on. Launching a game or game launcher (on a machine that has an OS) is not booting.