Hi Barry,
The widget_display_callback hook did, indeed, give me the answer I needed. For the sake of anyone else looking for a similar solution, I additionally used information on this page: http://wordpress.stackexchange.com/questions/57546/determining-a-widget-instance-and-sidebar-location
With that, I was able to determine which piece of information was going to be the most useful to me. I ended up using the $args[‘id’] value, and was able to include additional HTML in my list-widget template file based on that.
I did not need to keep the hook installed in my functions.php file in order for the IF statement in the template to keep working; I only needed it to find out what was in the various arrays.
Thanks again!