Let's assume that later versions of FM don't have any enhancements on this that I don't know about, and that there are no aftermarket or DIY API fixes.
What's the problem with having filenames with no extensions?
Framemaker (largely due to its Unix origins, and the utter lack of even suggested extension conventions until FM4 or so) doesn't care.
You could always create a "shortcut" to each file, using the same_name.fm
I just tested a shortcut 123456.fm linked to actual file 12345, and the "Filename (Short)" variable reports "123456".
Do the renaming from the Book menus.
The reverse, by the way, doesn't work (assembly of a book using no-extension shortcut files, linked to .fm files).
If you rename the book's component files so that they have no ".fm" file extensions, the system variable will work exactly the way you want, and the only "problem" that might result is that Mr.Bill's excuse for an operating system won't know what icon to display for the file, nor what to do if you double-click it (and if that matters, see the earlier suggestion to have parallel hard links or symlinks that do have the extension).
Looking at your example, where you show the 0009.fm appearing only in page headers and footers, there might be another approach. If the filename only appears on Master Pages, and is a fixed length in characters, put each instance, left-justified, in a text frame that is not wide enough for the ".fm" (and not tall enough, as it will wrap to the next line in that frame). The ".fm" will be there, but will not "appear". You can also accomplish the same trick by simply placing a masking rectangle on top of the .fm, set to page color (usually white) or as some named color set to "Cutout" in Color Views prior to render.