Contact Adobe Customer Service. They will need to release your serial number in order for you to use it on a future installation.
Your situation is not a purchased activated serial number trapped on a crashed computer. It is more a technical installation problem, rather than an activation problem - a fine line in this instance. And, you are not dealing with the current version of Premiere Elements which Adobe currently supports. So, you may be directed by Adobe Chat to seek help via Ask the Community which is us here in this forum.
But, it is worth the try to seek online Adobe Chat help at this time. For that purpose, please consider using the following link to online Adobe Chat.
If online Adobe Chat does refer you back to the Adobe Premiere Elements Forum (here), then I would consider a complete uninstall of the program, followed by a free ccleaner run through (regular and registry cleaner parts), reinstall with antivirus and firewalls disabled. No guarantee, but would be first start troubleshooting. Further details if needed.
Best wishes for success with online Adobe Chat. It is your ultimate solution if it recognizes the issue as activation rather than technical and will support a non current version. Please let us know the outcome as your schedule permits.