A few options:
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Installing in SteamOS directly - this should work, but will have to be redone every time steamOS updates. Some things like rwfus are meant to let you install things without them getting wiped, but that may or may not work. You can also create an installation script so you would just need to run a single file after a SteamOS update to restore everything.
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Try installing through nix. The deck has nix support, and nix has both mullvad client and the mullvad cli client.
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Just use the mullvad provided wireguard/openvpn files. In desktop they should be usable through the network manager, in game mode there's a Decky plugin called TunnelDeck.