If you have Adobe Illustrator here is what you need to do.
You need to save your file in Illustrator format.
Download the script for exporting the png files while keeping certain layer visible (it does exactly what you want but in Illustrator):
Then you need to add + to the background layer name (it will make your background appear in all your exported images -- you can read more about it in the link.)
Change all the layer names to something other than the default name.
Pick a location and you are done (for me it only works if i pick the location to be my desktop).
If you don't have Illustrator then you need to find/learn how to make/ask me to make/ a script that will do the same thing as the illustrator script does but for Photoshop.
You can try running the script with Photoshop by putting it in the Photoshop script folder and then going
but it did not work for me.
good luck :)