LR has a much better way of doing what you want.
Are you comparing it to Aperture?
It's called Virtual Copies.
Aperture provides the exact same functionality. In Aperture these are called Versions.
NB: each is a great and useful program. Neither was better, overall, than the other, except that now Adobe actively supports Lightroom and Apple abandoned Aperture.
If I make a Virtual Copy (which I do understand), I was under the impression that if I change the "File Name" on the Virtual Copy, it will also change on the File Name on the original. Is this correct?
Yes, that is correct.
Additionally, _all Virtual Copies must remain in the same folder as each other and as the Original on which they are based_.
There are a at least three things to consider as you work out a solution to your need.
1.
Copy Name. Each Photo (that is, each record in the Lightroom database) has, in addition to the file name with it shares with all Virtual Copies based on the same Original File, a Copy Name that is its own and independent of the Copy Names of all of the other Virtual Copies based on the same Original File. You can have the Copy Name show in the Metadata panel. This Copy Name is available to be used in the naming of files you create when you export a Photo from your Lightroom database. ("Export" means "create a file based on this Photo that I can share with other programs".) So you could assign the Copy Name "All Pixels set to Zero Saturation" to the Photo that has had all pixels set to zero saturation, and then use that Copy Name as part of the File Name of the File you create when you export that Virtual Copy Photo.
As with Aperture, it is helpful to wrap your brain around the fact that Virtual Copies (in Aperture, Versions) do not exist as image-format files until you create image-format files by exporting. (Lightroom misleads users who right-click a Virtual Copy and select "Show in Finder": it shows the user the file that was imported, which is almost always not the same image as the Virtual Copy. The context-sensitive menu item should be changed to "Show Original in Finder".)
2.
Storage Structure and Retrieval Structure. Folders — as in Finder folders, which is what Folders are in Lightroom (_not at all_ what Folders are in Aperture) — are of limited use for organizing many Photographer's output. You should see if using Collections (in Aperture, "Albums") better fits your work and workflow. As a general practice (for Lightroom and for Aperture), I suggest using a structure of Finder folders to _store_ your digital camera files, and create a structure of Collections in Lightroom (in Lightroom, Collections are grouped into Collection Sets, which are quite similar to what Folders are in Aperture) to _retrieve_ your work. At its simplest, put every shoot into its own folder, and put all output groupings into their own Collection.
3.
Duplicating Originals. Cave Canem: this is almost certainly _not_ a good practice. The easiest way I know to duplicate an Original is to right-click and select "Edit in Photoshop". If prompted, select one of the "edit a copy" options. In Photoshop, do nothing, but close the file and select "Save Changes". Back in Lightroom you will have two Photos, stacked. Each has it's own Original. You can drag either to any folder, and the other will stay put.
(Minor edit to first paragraph.)