This is a really good idea, but unfortunately I do not know how to code it
The save function is the only part of the SP-Studio I wasn't able to code myself. It was done by someone else and so I cannot change it. It saves the pictures as jpg files, so a transparent background is not possible. Even if I could change this to png files I don't think a transparent background would be possible, because it just makes a screenshot.
So unfortunately you need to do it this way:
Create a character with a one colored background (grey might be a good choice).
Save it or make a screenshot.
Load the picture into Photoshop / Gimp / a software like this.
Select the background with the Magic Wand.
Invert the Selection and you can copy the character to a picture of your choice.