Alright, so I probably can't help you with this without having some personal hands on experience with it.
What's probably happening is it's trying to install a system level systemd service, but you can't due to the steam deck being locked down. Your options would be to unlock the filesystem, find a way to install the Mullvad systemd service as a user instead, or figure out how to use systemd-sysext to install I as a system extension separate from the immutable filesystem.
Of those, I'm guessing the best solution would be the middle option, assuming you can get the right systemd mullvad-daemon.service file. Once you have it, it could be placed in ~/.config/systemd/user/ and enabled with systemctl --user enable mullvad-daemon.service. But as stated, the catch is you need the mullvad-daemon.service file, and I'm not sure the best way to do that. Maybe you could unlock the filesystem , let it install it as a system level service, and then convert it to a user level service? Either way it's complicated and I'd have to mess around with it myself to figure out what would work.
An alternative some people do is to use a smart plug to turn power off/on to the deck's dock (I'm assuming it's docked if you're using a controller).
There's a bios setting to wake up the deck whenever it's plugged in, so cycling power like that will wake up the deck.