SteamOS has been able to update the bios on the Steam Deck forever, I’m assuming since launch. The new patch is just for updating the bios when SteamOS is installed on other devices like the ROG Ally
Valve just open sourced SteamOS manager, which among other things allows updating the BIOS. I suspect we'll see this get added into Bazzite and other Linux distros meant for the Deck, but I don't think it's been added yet.
In the meantime, the Arch Wiki claims that fwupd works fine for updating the Steam Deck bios on Arch, and I'm assuming that means it would work for Bazzite and other distros as well.
Bazzite uses a "ujust" command to update everything (distro, flatpaks, distrobox containers, layered packages, etc.) and I'm pretty sure it includes bios updates. Unless it's just drivers that I'm thinking of..
If you click their "update system" icon, or just type "ujust update" in the terminal, it does the rest. Updates don't apply until you reboot, as is the case with all immutable distros (that I'm aware of).
Don't currently have access to double check though.
From looking at Bazzite documentation, ujust is a collection of scripts that run different updaters/etc. It does cover bios updates by running ujust enable-deck-bios-firmware-updates.
From looking at the Bazzite github, this runs a shell script that checks to make sure you have Steam Deck hardware and don't have the DeckHD replacement screen or 32GB ram mods installed, and then enables jupiter-biosupdate.service. I'm guessing that service is probably using fwupd for updates, but haven't been able to confirm it yet. If anyone else has some insight I'd be interested to hear it. I know the bazzite founder is active here but I'm going to restrain myself from pinging him and wasting his time over curiosity.