You can just copy the file and set XAUTHORITY as necessary. Just make sure only the desired user can read it.

No, do not do that. This gives access to the display to anyone who can connect to it. The proper way is to give the user access to file whose path is in $XAUTHORITY.

Capital letters in user names. 🤮

Debian has torbrowser-launcher you might wanna take a look at that.

As for the issue, this could be because the user lacks credentials to connect to the display.

Firstly, and most importantly, executing grub-install requires super-user privileges. Rather than adding it to PATH you should instead run the command through sudo. A regular user typically does not need any of sbin directories in their PATH.

As for the command itself, there are three things wrong with it:

1. PATH should only include directories whereas you tried to add to it a path to an executable. So rather than /usr/sbin/grub-install/grub-install you should just add /usr/sbin.
2. White space is significant, so the space before colon would make your command not work anyway.
3. Rather than appending to PATH you’ve overwritten the variable. Instead you need PATH="$PATH:/usr/sbin/:/usr/local/sbin" (notice $PATH: at the beginning of the assignment).

Also, export is unnecessary since PATH is already an environment variable. (That’s also bashism but that’s likely an irrelevant issue).