The toolbar itself, and its configuration, is a piece of macOS technology. How it is stored and loaded is entirely dictated by the system, rather than something that can be safely decoded and coded into a separate file (at least in a long-term sense, as Apple may change how it is saved from one year to the next).
You could transplant the ~/Library/Preferences/com.literatureandlatte.scrivener3.plist file from one computer to another, of course, with Scrivener closed. That’s probably the best way, as the structure used to define the toolbar configuration looks a bit complex, and may not be something you could write and read out individually, via the defaults command-line tool.