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 https://wiki.archlinux.org/title/MangoHud.
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