Document Analyzer does extract the script to script.qvs in the metadata directory. Problem here is that DocAnalyzer needs to open the qvw so same PE limitation would apply.
Do you have document logs you can recover the script from? Can you use the recovery option (you should get the option to recover the document and update with a new PE license 4 times).
I cannot really test this because of lacking a PE Desktop. However, I think it should work for you as well. It works with my documents.
- Install Qlik Sense
- Throw one of your lost QVW documents on the Desktop-hub screen
- Open the Editor for loading data (button at the top-left, second menu entry)
- Cop'n'Paste the script to a text editor like notepad(++)
- Save the script to a .qvs file
Note that a PE desktop isn't really meant for continued development of multiple documents in a business environment. It really is a Try-out version. You will run into trouble again. Consider buying a license.