For googlers who find this thread down the road:
I made a tool to batch stamp PDF filenames. You can get it here:
http://saintjohnny.com/pdffilenamelabeler
I've also added so that you can also stamp prefix/suffix before and after the filename, as well as stamp custom text ignoring filenames, and change the height of the stamp to go on the top of the page as a header if needed. Since this isn't a true footer/header, it retains the information when combining PDFs later on which is a plus.