You need a clipping path like in Photoshop. This is possible with the Elements+ affordable add-on. I'll come back tomorrow when I am on my computer.
Your detailed description and the link to YouTube show that your printer driver works very differently from conventional inkjet printers. It automatically fills the background (matte???) with white, so that the only suggested process is to include a 'clipping path' in your image.
Since Elements does not have the full 'path' panel, you need an additional script to restore that functionality. My suggestion is to download the very affordable Elements+.
About Elements+
Look at the 'paths' section.
There is a script to create a path from a selection you have made in Elements (same selection way as in Photoshop):
Creating Path from Selection
Once you have created this 'clipping path', you are in the same situation as in Photoshop. You should be able to print directly from Elements or from a saved copy in psd, tiff... or even jpg format.
Yes, the jpeg format can't store layers, but it can store a clipping path.
I am not sure you can retrieve it from the forum, here is the DropBox link:
Dropbox - ClippingPath.jpg
