Okay, Skwire, the renaming approach worked.
Chris, For some reason SubCopy doesn't allow directly inputting or even editing the field "Source file." (Only option is Windows navigation and selecting.) If user could do that, it may be possible to place the full path and name of the shortcut file.
That would be easier than having to rename, copy to subfolders, and then recursively rename back again.
But, hey, at least the task can be accomplished. I'm quite thankful for that.