EPIC released Fall Guys for free on their store, but removed it from Steam so this is the only available version to get for now. The issue is that the Steam version is easy to play on, but the EPIC version needs some work. Gathering some info on the internet, here are the steps I followed to get it working.
Source: https://www.reddit.com/r/linux_gaming/comments/vi54sh/eos_overlay_on_egs_fall_guys/ and https://old.reddit.com/r/SteamDeck/comments/vht8d1/fall_guys_f2p_epic_version_tutorial_for_steam_deck/
//EDIT: this is not relevant anymore as there are methods since Heroic Games Launcher 2.4.0 to have everything to work without any hassle (tick the box for EasyAntiCheat Runtime and play), no need for Steam.
Install Fall Guys through the Heroic Games Launcher. Make sure you have set a capable WINE version in Heroic or Fall Guys Settings, for example ProtonGE 7.20 or other capable WINE version. Also disable in Heroic Settings or Fall Guys Settings things you prefer to control through Steam Launch Options (mangohud, gamemode, and so on..).
Make a symlink or a copy of <FallGuys>/EasyAntiCheat/easyanticheat_x64.so to folder <FallGuys>/FallGuys_client_game_Data/Plugins/x86_64/ then edit <FallGuys>/FallGuys_client.ini and change TargetApplicationPath to FallGuys_client_game.exe
After that there are basically two main steps, getting the overlay to work to be able to invite friends or join their lobby, and getting the anti cheat to work to be able to start a game.
Get the EPIC Online Services to work:
In Fall Guys Settings in the Heroic Games Launcher do the following:
Start the game from Heroic once, wait for the Play button to change status then return to normal.
Remove the Alternative Executable and Game Arguments from Fall Guys Settings.
Start the game from Heroic again, a popup for updating EOS should download large update, once done the game should start, with a working EOS overlay. At this point still no working EAC.
Get Easy Anti Cheat to work with Proton:
Run the Heroic Bash Launcher to generate Fall Guys Heroic startup script for Steam, following the documentation (run setup.sh to create the startup scripts, then AddToSteam.sh to select and add Fall Guys to Steam, then start Steam)
Install Proton EAC Runtime steam://install/1826330 in Steam
Change Launch Options in Steam for Fall Guys to PROTON_EAC_RUNTIME="/home/omano/.steam/steam/steamapps/common/Proton EasyAntiCheat Runtime/" %command% (adjust to proper installation path) and do not force another Proton version in Compatibility tab
Game should now be able to start from Steam and have a working EAC, and EOS overlay, to be able to play with friends.
Unlike the Steam version, there is no pre-compiled shaders automatically downloading, so the game will have huge lags/freezes until the shaders are all compiled from playing the various game environments. But at least everything should work from these steps.