Most linux OS with locked down file systems allow you to install apps permanently if you need something that doesn’t work with flatpak/distrobox/etc. It’s usually called something like atomic updates, and it installs the software in a layered image, and will require a reboot to load.

It’s not the ideal way to install software, but it’s a lot easier than your options for installing packages on SteamOS.

I haven’t personally used Bazzite, so I’m not 100% sure that it supports it, but I’m assuming it does. If anyone else wants to confirm it I would appreciate it.