Instead of trying to get Steam to write outside of its sandbox, I would configure the OS to search Steam's files for application launchers.

Your system searches "data" directories for directories called applications, and loads .desktop files from there. You can customize locations for data directories using the XDG_DATA_DIRS environment variable. See https://unix.stackexchange.com/a/492878

In this case you want to set XDG_DATA_DIRS to include ~/.var/app/com.valvesoftware.Steam/.local/share because that is the parent of the applications directory.

I think you'll want to include the default data dirs so you don't lose your other launchers. So something like,

export XDG_DATA_DIRS="/usr/local/share/:/usr/share/:$HOME/.var/app/com.valvesoftware.Steam/.local/share"

If I'm understanding the spec correctly you don't need to include ~/.local/share because that is the default path for XDG_DATA_HOME which is always searched, and XDG_DATA_DIRS specifies additional paths to search.

I believe its an option when you right click the game in steam.

If you don't want the desktop icons, you don't need to relax the flatpak permissions for the xdg-desktop. Just the first two for allowing steam to see the icons and the applications folders.

If the right click option isn't there in steam, there might be a way to regenerate the icon, but off the top of my head I'm not sure. Reinstalling the game should do it but that's a pain for larger games.