Hey chmsant,
This will seem counterintuitve, but some have actually had success restoring 9.1 backups to devices running 9.0.1. You try putting the phone into Recovery Mode and connecting it to iTunes, allowing it to restore you back to the latest stable version of iOS (9.0.1). Then try the backup again from there.
Failing that, you could try a reset to "factory condition" (I appreciate that your phone is effectively brand new) as this is one of Apple's suggested troubleshooting steps in cases like this. It's Settings > General > Reset > Erase all Content and Settings, and then try again from the same iTunes backup.
Given that this is happening to you, it will probably happen to others too and should be fixed in future version of iOS 9.1 and/or iTunes 12. In the meantime, you could speed this on a little by submitting a bug report through the Feedback Assistant app that will be on your phone running 9.1 or at https://bugreport.apple.com
Finally, if you're prepared to spring for one of them, there are at least two 3rd party softwares that will probably succeed to restore the backup to your iPhone.
Max.