You can do it the other way around, though only one photo at a time. Open Photos, select a photo, then share it. You can send it to Dropbox
Or if it is something you do a lot you can probably set up something in the Workflow app. It's a simple two step workflow. Took me all of a minute to set up and test. However, I no longer have Photo Stream turned on, so I can't guarantee it will work there. It does work with iCloud Library (beta) and iCloud Photo Sharing.
I'll be happy to walk you through the setup for this if you decide to use it. Or just publish and email you a download link for the workflow.
Edit: A little more testing shows that those photos that I copied from iCloud Photo Sharing were of low quality. Opening the photos in the Photos app first, so that the full resolution photo was downloaded, corrected this. No idea how it would work with Photo Stream or iCloud Photo Library if they don't have full version local to the iPad. That can be optional in the second case.