Ummm... how to do this? I'm stumped...

I have a file that I would like to print like so:

The swan layer is set to "darken" transparency style in the layers panel. This works when I have this brown background behind it.

But, I need to delete this brown background so that my client can print on actual kraft paper. When I delete the background, my swan layer no longer has any layer behind it to "darken" into, so I get this resulting file, where the swan is colored in with white:

It would be pretty hard to get the white out of the swan file, since it is hand-drawn... Is there another way to allow for the swan to print without any white? There's got to be some sorta trick that I'm not aware of...

Thanks!