I think, unfortunately, you might be out of luck. I don’t think you can migrate data to a new 3DS that way – even with the same NNID.

You could try putting a blank SD card in your new 3DS, initialising it, then transferring the data from your backup onto it into the new folders created by the initialisation as per my guide. If that doesn’t work, I don’t think there’s any hope unless Nintendo can suggest something.

I had the same thing on my old Wii when it was replaced by Nintendo. All my games could be redownloaded, but my saves were gone forever – including my Animal Crossing one 🙁