Thanks for the cool idea, but this will not be possible.
If the BG was alpha transparent, it would only show what was behind the BG and putting something behind the BG Layer is not possible, because it is the last layer and cannot be moved. However, you should already be able to do this if your video mixer supports Chroma Key. This will replace the Black color of the Stage with the source from your video mixer.
The article in this LINK may have information you might find useful.
Hope this helps.