What version of Premiere Elements on what computer operating system?
I know that you as the account holder can download as .mp4 your video uploaded to YouTube, but I do not know if you can do that sort of thing with Facebook.
But, here on the Premiere Elements side of things is not the place to find the confirming answer.
Please consider posting your question in the Facebook Forum. The following is a sampling of what I found online
Please let us know the outcome as your schedule permits.