I want to remove the shortcut bar on only one desktop icon--of the Firefox browser.
I know there's the TweakUI program, but I'm pretty sure that program makes ALL desktop icons have no shortcut bar, right? Obviously, I do not want that.
Is there a registry hack that would accomplish this?
No, you can only trigger it for all icons because the shortcut icon is just an overlay for the base program icon and it is superimposed on the main icon.
You can disable showing the arrows on icons by deleting the IsShortcut parameter like it is shown here or tweaking the icon mapping and redirecting calls for arrow to the transparent icon: