Thank you, DaveRI. I tried that, and it seemed to work for the folder layout. The first article you cited, "How do I use snapshots in Fences?", leaves a lot for the reader to figure out on his own--like that there are three files to be copied over for each one snapshot, and that the files are not named according to their titles as displayed in "Layout and snapping". So for instance, the snapshot I created was displayed as "11/16/2015". But down in the folder the name of the image file was 4503887e-27a0-4fa1-9788-dcab2b1018b8.png. The creation date/time was helpful of course, but I felt more sure it was the correct snapshot when I checked the XML file of the same name, which contained a Title tag with the value "11/16/2015". So I copied the png file, the xml file, and a png file of the same name with "_thumb" appended, none of which is mentioned in the article.
What that action didn't copy over is the rules and settings. A full configuration export/import would have those as well. So I did a comparison of values for the keys under CURRENT_USER\Software\Stardock\Fences. My desktops aren't identical, so I didn't want to copy Icons. There didn't seem to be a need to duplicate InitialSnapshot, Skinning or Tips. So I just exported Groups, Rules and Settings out of one computer and imported them into the other. As far as I can tell, it did what I wanted. When I went to "Sorting / Organizing" I found the correct rules present and checked.
But now the icons are mentioned in the XML file and not in the registry. Those icons don't exist on my desktop anyway, of course. So I have some data corruption, in a sense. It doesn't seem to be hurting the Fences program so far, but it's unsettling just the same. I wonder how the desktop would have looked if I had copied over the "Icons" section of the registry and not the snapshot.