If you turn the top layer with the design into a smart layer then manually update the layer comps once and save the file as a template you can then use the menu item Layer-Smart Object-Replace Cotents... to change out the design in a way that would no longer require updating the layer comps. Changing the content of a smart object does not effect the layer comp status like adding a new layer would.
In the Layer Comps to Files.jsx if you find the line
var fileNameBody = exportInfo.fileNamePrefix;
and replace it with
var fileNameBody = app.activeDocument.layers.name;// top layer name
The script will no longer use the prefix from the dialog and instead use the top layer's name as the prefix.
To automate the whole process would require creating a custom script. The Layer Comps to Files script was not written to be called from another script.
THats great, thank you very much for your help.
I may look at the fully automated script - please can you let me know your email address if you could write such a script?
Hey I came across this with a similar question. How can I make it the document name instead of the top layer name? I know this is old news, but I'd appreciate it!