I don't use flatpak but I didn't have to configure Lutris or Steam in any way for that.

If you look around goverlay you'll find that there are two ENVs that you can set up in X/wayland service to have MangoHud automatically in every context it can render in.

So first try running with MANGOHUD=1 env exported. If that doesn't help, try mangohud steam.
If I understand correctly, flatpack run in some kind of container, so it's possible that you might need to set the env or the command, so it happens inside the container

Flatpak apps don't have access to your system packages, so you need to install mangohud as flatpak. Once it's installed it's available to Steam flatpak and can be enabled like system mangohud in system Steam.

https://github.com/flightlessmango/MangoHud?tab=readme-ov-file#flatpak

Edit: Switching from system Steam to flatpak Steam is simple and it's always possible to switch between them. Just make sure to give flatpak Steam access to the existing SteamLibrary through flatseal.

Personally I have my SteamLibrary at ~/Games/SteamLibrary and give flatpak Lutris/Steam access to ~/Games.

I tried so hard to use the Steam Flatpak, but hit a wall when I wanted game libraries on multiple drives. The Arch wiki recommended to use the native binary.

I too want a HUD.