So I managed to solve it. After searching around, many posts pointed to removing the xdg-desktop-portal-gnome package, which would help for other issues as well. This didn't work for me though.
It turned out I had to remove the ~/.local/share/flatpak/db/documents file. After that I could copy files again. Now on to the next issue where the linux-surface kernel doesn't get signed with my own MOK after the kernel gets updated.