Follow up, have both the Flatpak version and package steam up and running. Moved my game library to ~/Games/Steam. I added ~/Games/Steam:rw (and later ~/Games/Steam:create) to my Flatpak permissions and tried to install a game that already existed to make Flatpak Steam realize it was there, Steam instead gives me a “Disk Write Error”. Did you hit this at all, any idea what it may be?
EDIT: Fix it, or it magically fixed itself. I removed the Steam library and re-added it and that seemed to make it happy.