Answer
RMB on the relevant Navigator entry and go to Properties — there's an 'Own Process' checkbox there; checking it auto-assigns a Process ID (though you can specify your own preferred string instead of the auto-assigned one). Use that same Process ID value in your Custom Menu configuration, and the menu item will then only appear in windows opened through that specific navigator entry/process, rather than in every window that happens to use the same LU. When a page opened with an own process is active, you can see the process ID appear as a parameter in the browser's address bar. There's no central screen listing every existing process ID in the environment, so keeping your own naming record is worthwhile.