Can you get EDDiscovery running on the Deck?
If you can get it running, then I'd dock the Deck (I'm a poet) and set up EDDiscovery in desktop mode so that the overlay renders over the game window as you want. You could then make an executable bash script that launches both at the same time, although how you would get that to run in game mode rather than desktop mode is something I'd have to work out vicariously; I don't own a Deck.
You could also try installing the Windows version of EDDiscovery in the same Proton prefix as the game.
It looks like you can build for Linux from source using Mono. This is with the caveat that it looks really buggy.
An alternative would be to run it through Wine (just like Elite runs through Proton). If you don't want to configure Wine manually, you could look into using
Bottles to install it - it gives you a straightforward GUI for Wine, including options to install common Wine dependencies for programs (like dotnet, which is what I'm guessing Mono is standing in for in the Linux build). Bottles is only available as a Flatpak, so you'll need to give Flatpak permissions to access files outside its sandbox (only do this if you trust Bottles and EDDiscovery). If you don't fancy doing
that manually, you can install
Flatseal, a permissions manager for Flatpak.
I've used Bottles to install EDHM_UI on Linux, and it worked. It took a bit of bashing my head off the keyboard to work out how to do it, but it did work.
You could also ask this question on the
EDDiscovery Discord - I'd be surprised if they haven't had this question before.
P.S. Won't the overlay be
really tiny on the Deck's screen?