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.
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.